2.1 Conformal regular Cantor set

A $C^m$ regular Cantor set (or dynamically defined Cantor set) on the complex plane is given by the following data: a finite set $\mathbb {A}$ of letters and a set $B\subset \mathbb {A}\times \mathbb {A}$ of admissible pairs, for each $a\in \mathbb {A}$ a compact connected set $G(a)\subset \mathbb {C}$ , and a $C^m$ function $g:V\to \mathbb {C}$ defined in an open neighborhood V of $\bigsqcup _{a\in \mathbb {A}} G(a)$ . This data must verify the following assumptions.

• • The sets $G(a)$ , $a\in \mathbb {A}$ , are pairwise disjoint.

• • $(a,b)\in B$ implies $G(b)\subset g(G(a))$ , otherwise $G(b)\cap g(G(a))=\emptyset $ .

• • For each $a\in \mathbb {A}$ , the restriction $g|_{G(a)}$ can be extended to a $C^m$ diffeomorphism from an open neighborhood of $G(a)$ onto its image such that $m(Dg)>1$ (where $m(A)=\inf _{v\neq 0} {|Av|}/{|v|}$ is the minimum norm of the linear map A).

• • The subshift $(\Sigma ^+, \sigma )$ induced by B

  $$ \begin{align*}\Sigma^+=\{\underline{a}=(a_0,a_1,\ldots)\in \mathbb{A}^{\mathbb{N}}:(a_i,a_{i+1})\in B \text{ for all } i\geq 0\},\end{align*} $$
  $\sigma (a_0,a_1,a_2,\ldots )=(a_1,a_2,\ldots )$ is topologically mixing.

Once we have such data, we can define a Cantor set (that is, totally disconnected, perfect compact set) on the complex plane $K=\bigcap _{n\geq 0}g^{-n}(\bigsqcup _{a\in \mathbb {A}} G(a))$ .

We will say that the regular Cantor set is conformal if for all $x\in K$ , the linear map $Dg(x):\mathbb {R}^2\to \mathbb {R}^2$ is conformal, that is, $m(Dg(x))=\|Dg(x)\|$ . The assumption that $Dg$ is not necessarily conformal outside of the Cantor set was introduced by Araujo in his PhD thesis, where he studied Cantor sets associated to complex horseshoes for automorphisms of $\mathbb {C}^2$ . It plays an important role in the investigation of a complex version of Palis’s conjecture. We will write only K to represent all the data that are required to define a conformal regular Cantor set. All Cantor sets in this paper will be conformal regular Cantor sets; we will usually refer to them just as Cantor sets.

The degree of differentiability, m, can be any real number bigger than one. If m is not an integer, then g being $C^m$ means that it is $C^{[m]}$ , where $[m]$ is the integer part of m, and $D^{[m]}g$ is Holder with exponent $m-[m]$ . To prove our results, we will assume that $m\geq 2$ .

We can actually suppose that the sets $G(a)$ verify $G(a)=\overline {\mathrm {int}(G(a))}$ , this is a consequence of the next lemma.

2.2 Limit geometry

Associated to K, we define the sets $\Sigma ^\textit{\,fin}=\{(a_0,\ldots ,a_n):(a_i,a_{i+1})\in B \}$ and $\Sigma ^{-}=\{(\ldots ,a_{-n},\ldots ,a_{-1},a_0): (a_i,a_{i+1})\in B\}$ .

Given $\underline {a}=(a_0,\ldots ,a_n)$ , $\underline {b}=(b_0,\ldots ,b_m)$ , $\underline {\theta }^1=(\ldots ,\theta ^1_{-1},\theta ^1_0)$ , $\underline {\theta }^2=(\ldots ,\theta ^2_{-1},\theta ^2_0)$ , we will use the following notation.

• • If $a_n=b_0$ , $\underline {a}\underline {b}=(a_0,\ldots ,a_n,b_1\ldots ,b_m)$ .

• • If $\theta ^1_0=a_0$ , $\underline {\theta }^1\underline {a}=(\ldots ,\theta ^1_{-1},\theta ^1_0,a_1,\ldots ,a_n)$ .

• • If $\theta ^1_0=\theta ^2_0$ , $\underline {\theta }^1\wedge \underline {\theta }^2= (\theta ^1_{-j},\ldots ,\theta ^1_{0})$ , where j is such that $\theta ^1_{-i}=\theta ^2_{-i}$ for all $0\leq i\leq j$ , and $\theta ^1_{-j-1}\neq \theta ^2_{-j-1}$ .

• • If $\theta ^1_0=a_n$ , $\underline {\theta }^1\wedge \underline {a}= (\theta ^1_{-j},\ldots ,\theta ^1_{0})$ , where j is such that $\theta ^1_{-i}=a_{n-i}$ for all $0\leq i\leq j$ , and $\theta ^1_{-j-1}\neq a_{n-j-1}$ .

For $\underline {a}=(a_0,\ldots ,a_n)\in \Sigma ^\textit{\,fin}$ , define $G(\underline {a})=\{x\in \bigsqcup _{a\in \mathbb {A}} G(a) : g^j(x)\in G(a_j), j=0, 1,\ldots ,n\},$ and the function $f_{\underline {a}}:G(a_n) \to G(\underline {a})$ given by

$$ \begin{align*}f_{\underline{a}}=g|_{G(a_0)}^{-1}\circ \cdots \circ g|_{G(a_{n-1})}^{-1}.\end{align*} $$

Denote by $K(\underline {a})$ the set $K\cap G(\underline {a})$ . For each $a\in \mathbb {A}$ , we choose an arbitrary point $c_a \in K(a)$ . Using this, define $c_{\underline {a}}\in G(\underline {a})$ by $c_{\underline {a}}=f_{\underline {a}}(c_{a_n})$ .

Notice that $Df_{\underline {a}}(c_{a_n})$ is a conformal matrix in $\mathbb {R}^2$ , then it is equal to a positive real number times a rotation matrix, and denote the angle of rotation by $v_{\underline {a}}\in \mathbb {R}/(2\pi \mathbb {Z}).$ In this way, we have a preferred point and direction for each $G(\underline {a})$ . We also define $r_{\underline {a}}=\mathrm {diam}(G(\underline {a}))$ , where $\mathrm {diam}$ means diameter.

Given $\underline {\theta }=(\ldots ,\theta _{-n},\ldots ,\theta _{-1},\theta _{0})\in \Sigma ^{-}$ , let $\underline {\theta }^n=(\theta _{-n},\ldots ,\theta _0)$ and define $k^{\underline {\theta }}_n:G(\theta _0)\to \mathbb {C}$ by

$$ \begin{align*}k^{\underline{\theta}}_n=\phi_{\underline{\theta}^n}\circ f_{\underline{\theta}^n},\end{align*} $$

where $\phi _{\underline {\theta }^n}$ is the unique map in $Aff(\mathbb {C})=\{A(z)=az+b:a,b\in \mathbb {C},\,\,a\neq 0\}$ such that $\phi _{\underline {\theta }^n}(c_{\underline {\theta }^n})=0$ , $D\phi _{\underline {\theta }^n}(c_{\underline {\theta }^n})e^{iv_{\underline {\theta }^n}}\in \mathbb {R}^+$ , $\mathrm {diam}(\phi _{\underline {\theta }^n}(G(\underline {\theta }^n)))=1$ . For the next theorem, we consider $k^{\underline {\theta }}_n$ extended to a small open neighborhood $G^*(\theta _0)$ of $G(\theta _0)$ (as in Lemma 2.1).

Theorem 2.1. Let K be a $C^m$ conformal Cantor set. For any $\underline {\theta }\in \Sigma ^-$ , the family of functions $k^{\underline {\theta }}_n:G^*(\theta _0)\to \mathbb {C}$ converges in the $C^{[m]}$ topology, with an exponential rate of convergence independent of $\underline {\theta }$ , to a $C^m$ function $k^{\underline {\theta }}:G^*(\theta _0)\to \mathbb {C}$ . The function $k^{\underline {\theta }}$ is a diffeomorphism onto its image and the derivative $Dk^{\underline {\theta }}(x)$ is conformal for all $x\in K(\theta _0)$ .

Moreover, if $m\geq 2$ , then there is a constant $C>0$ such that given $\underline {\theta }^1, \underline {\theta }^2 \in \Sigma ^-$ ending with the same letter,

$$ \begin{align*} \sup_z [|k^{\underline{\theta}^1}\circ (k^{\underline{\theta}^2})^{-1}(z)-z|+ \|D(k^{\underline{\theta}^1}\circ (k^{\underline{\theta}^2})^{-1})(z)-I\|]\leq C {\mathrm{diam}} (G(\underline{\theta}^1\wedge \underline{\theta}^2)). \end{align*} $$

The function $k^{\underline {\theta }}$ is called a limit geometry of K. Notice that the rate of convergence being independent of $\underline {\theta }$ implies that $D^l k^{\underline {\theta }}$ , for $0\leq l\leq [m]$ , depends continuously on $\underline {\theta }$ . The proof of Theorem 2.1 can also be found in [Reference Araújo and Moreira1].

For $\underline {\theta } \in \Sigma ^{-}$ , $\underline {a}\in \Sigma ^\textit{fin}$ , such that $\underline {a}$ starts with the last letter of $\underline {\theta }$ , define

$$ \begin{align*} &G^{\underline{\theta}}(\underline{a})=k^{\underline{\theta}}(G(\underline{a})),\quad K^{\underline{\theta}}(\underline{a})=k^{\underline{\theta}}(K(\underline{a})),\quad c^{\underline{\theta}}_{\underline{a}}=k^{\underline{\theta}}(c_{\underline{a}})\\ &\text{exp} (iv^{\underline{\theta}}_{\underline{a}})=\frac{Dk^{\underline{\theta}}(c_{\underline{a}})}{\|Dk^{\underline{\theta}}(c_{\underline{a}})\|} \text{exp}(iv_{\underline{a}}),\quad r^{\underline{\theta}}_{\underline{a}}=\mathrm{diam}(G^{\underline{\theta}}(\underline{a})). \end{align*} $$

Let $F_{\underline {a}}^{\underline {\theta }}$ be the affine map determined by the equation $k^{\underline {\theta }}\circ f_{\underline {a}}=F_{\underline {a}}^{\underline {\theta }}\circ k^{\underline {\theta }\underline {a}}$ . It maps $0$ to $c_{\underline {a}}^{\underline {\theta }}$ and can be written using $r_{\underline {a}}^{\underline {\theta }}\in \mathbb {R}^+$ , $v_{\underline {a}}^{\underline {\theta }}\in \mathbb {R}/2\pi \mathbb {Z}$ as

$$ \begin{align*}F_{\underline{a}}^{\underline{\theta}}(z)=r_{\underline{a}}^{\underline{\theta}} \text{exp}(iv_{\underline{a}}^{\underline{\theta}})z+c_{\underline{a}}^{\underline{\theta}}.\end{align*} $$

2.3 Mass distribution principle

Typically, estimating the Hausdorff dimension from below is harder than from above. One usual technique is the mass distribution principle that we state below.

The next proposition is a consequence of the mass distribution principle and it will be used to prove the desired dimension formula (Theorem B). Its proof is not difficult and can be found in [Reference Zamudio13, §1.3].

Let N be the node set of a rooted tree with the property that every node has finite index. Here, N can be described in the following way: there is a marked element $p_0\in N$ called the root of N; for each $p\in N$ , we have a finite set $Ch(p)\subset N$ called the children of p; if $p\neq q$ , then $Ch(p)\cap Ch(q)=\emptyset $ ; for any $q\in N$ , there is a sequence $q_0,q_1,\ldots ,q_m$ such that $q_0=p_0$ , $q_m=q$ , and $q_{i+1}\in Ch(q_i)$ , $i=0,\ldots ,m-1$ , such q is called an m-level node of N. Denote by $I(k)$ the set of k-level nodes. Now N can be written as the disjoint union $N=\sqcup _{k=0}^{\infty } I(k)$ .

2.4 Not essentially real Cantor sets

In this subsection, we will present a hypothesis in the Cantor set that will guarantee it is indeed two dimensional. We remark that for regular Cantor sets in the real line, there is no analogous definition and results to those in this section. Those are objects that start appearing at dimension two.

It is not difficult to prove that K is essentially real if and only if for every $\underline {\theta }\in \Sigma ^-$ , the limit Cantor set $K^{\underline {\theta }}(\theta _0)$ is contained in a straight line. Moreover, one can prove that K being essentially real is equivalent to K being contained in a $C^1$ one-dimensional manifold embedded in the plane. For the proof of the scale recurrence lemma, we will suppose that the Cantor set is not essentially real, in such a case, one is able to control the quantity of elements $G^{\underline {\theta }}(\underline {a})$ close to an arbitrary line; this is the content of the next lemma.

Given $c>0,\rho >0$ , define $\Sigma (c,\rho )=\{\underline {a}\in \Sigma ^\textit{fin}:\,\,c^{-1}\rho \leq \mathrm {diam}(G(\underline {a}))\leq c \rho \}$ . We can think of this as the set of $G(\underline {a})$ having approximate size $\rho $ . Using standard techniques (see [Reference Palis and Takens10] or [Reference Zamudio13]), one can prove that there is a constant $C>0$ , depending only on c and the Cantor set K and not depending on $\rho $ , such that

$$ \begin{align*}C^{-1} \rho^{-HD(K)} \leq \#\Sigma(c,\rho)\leq C \rho^{-HD(K)}.\end{align*} $$

Suppose we have fixed a constant $C_5>0$ . Let $(a,b)\in B$ . A subset $D\subset \Sigma (C_5,\rho )$ is called a discretization of $K(a,b)$ of order $\rho $ if $\bigcup _{\underline {a}\in D}K(\underline {a})=K(a,b)$ .

Lemma 2.2. Let K be a Cantor set not essentially real. There exist an angle $\alpha \in (0,\pi /2)$ and numbers $\rho _2>0$ , $a\in (0,1)$ , depending only on $C_5$ and the Cantor set K, such that for any limit geometry $k^{\underline {\theta }}$ , $x\in G^{\underline {\theta }}(\theta _0)$ , line L, $s\in \mathbb {A}$ , D discretization of $K(\theta _0, s)$ of order less than $\rho _2$ ,

$$ \begin{align*}\#\{\underline{a}\in D:\,\, G^{\underline{\theta}}(\underline{a})\cap \mathrm{Cone}(x,L,\alpha)\neq \emptyset \}\leq a\cdot \#D,\end{align*} $$

where $\mathrm {Cone}(x,L,\alpha )$ is the set of $z\in \mathbb {C}$ such that the vector $z-x$ forms an angle of measure less than $\alpha $ with the line L.

Another use of the not essentially real hypothesis will be given in the next lemma. Let K be a Cantor set, for $x \in K$ , consider the set

$$ \begin{align*}K^{dir}_x :=\bigcap_{\delta>0} \overline{\bigg\{\frac{y-x}{|y-x|}:y\in B_{\delta}(x)\cap (K\setminus \{x\})\bigg\}},\end{align*} $$

where $B_{\delta }(x)$ is the open ball of radius $\delta $ centered at x. If K is not essentially real, then the set $K^{dir}_x$ has two linearly independent vectors (over $\mathbb {R}$ ) and then the following lemma holds for K.

Lemma 2.3. Let K be a Cantor set and f a $C^l$ function from a neighborhood of K into $\mathbb {R}^2$ . Suppose that f is conformal at K, that is, $Df(x)$ is conformal for all $x\in K$ , and $K^{dir}_x$ has two linearly independent vectors (over $\mathbb {R}$ ) for all $x\in K$ . Then, for all $x\in K$ , the l-linear map $D^l f(x):\mathbb {R}^2\times \cdots \times \mathbb {R}^2 \to \mathbb {R}^2$ is conformal, that is, there is a complex number $c^l_x$ such that

$$ \begin{align*}D^l f(x)(z_1,\ldots,z_l)=c^l_x\cdot z_1 \cdot z_2 \cdot \cdots \cdot z_l.\end{align*} $$

The operation $\cdot $ on the right-hand side of the last equality corresponds to complex multiplication.

In particular, if a Cantor set is not essentially real and not essentially affine, then for the values $z_0 \in K$ , $\underline {\theta }^1,\underline {\theta }^2\in \Sigma ^{-}$ , given by Definition 2.1, there is a non-zero complex number $d_0$ such that

$$ \begin{align*}D^2(k^{\underline{\theta}^1}\circ (k^{\underline{\theta}^2})^{-1})(z_0)(v,w)=d_0\cdot v \cdot w.\end{align*} $$

2.5 Renormalization operator

From now on, we will be working with a finite set of conformal regular Cantor sets $K_1,\ldots ,K_n$ . To each of them, we have various objects associated, as defined in the previous subsections. We will use subscripts and superscripts to differentiate the objects from one Cantor set to the other. For example, we use $\Sigma _j(c,\rho )$ for the set $\Sigma (c,\rho )$ , which was defined in the last subsection, associated to the Cantor set $K_j$ . We will denote by $d_j$ the Hausdorff dimension of the Cantor set $K_j$ . In this section, we will define renormalization operators, which are operators associated to the family $K_1,\ldots ,K_n$ of Cantor sets.

Define $J=\mathbb {R}^{n-1}\times \mathbb {T}^n$ , where $\mathbb {T}=\mathbb {R}/(2\pi \mathbb {Z})$ . The group $\mathbb {T}$ is endowed with the unique invariant distance giving diameter $2\pi $ to the whole space. For $v\in \mathbb {T}$ , we denote by $\|v\|$ the distance between v and the zero element. The space J is an abelian locally compact group. We put on J the unique invariant metric such that the distance between $(t_1,\ldots ,t_{n-1},v_1,\ldots ,v_n)$ and the zero element is $\max _j\{|t_j|,\|v_j\|\}$ .

Lemma 3.1 justifies why we choose to work on $J=\mathbb {R}^{n-1}\times \mathbb {T}^n$ . The lemma states that the ‘renormalization’ of h, given by $h\circ (f_{\underline {a}^1},\ldots ,f_{\underline {a}^n})$ , becomes, modulo composition by an affine function, arbitrarily close to a function of the form $B\circ A(t,v)\circ (k^{\underline {\theta }^1},\ldots ,k^{\underline {\theta }^n})$ . The pair $(t,v)$ , parameterizing the maps $A(\cdot ,\cdot )$ , is in $\mathbb {R}^{n-1}\times \mathbb {T}^n$ . In fact, if the function h were not only $C^1$ but holomorphic, then we could reduce further the type of function A appearing and take J as $\mathbb {R}^{n-1}\times \mathbb {T}^{n-1}$ . The renormalization operators are then defined observing how the expression $A(t,v)\circ (k^{\underline {\theta }^1},\ldots ,k^{\underline {\theta }^n})$ changes when one composes it by $(f_{\underline {b}^1},\ldots ,f_{\underline {b}^n})$ .

For $(\underline {b}^1,\ldots ,\underline {b}^n)\in \Sigma ^\textit{fin}_1\times \cdots \times \Sigma ^\textit{fin}_n$ , define the operator

$$ \begin{align*} T_{\underline{b}^1,\ldots,\underline{b}^n}:\Sigma^-_1\times \cdots\times \Sigma^-_n \times J \to \Sigma^-_1\times \cdots\times \Sigma^-_n \times J, \end{align*} $$

given by

$$ \begin{align*} &T_{\underline{b}^1,\ldots,\underline{b}^n}(\underline{\theta}^1,\ldots,\underline{\theta}^n,t_1,\ldots,t_{n-1},v_1,\ldots,v_n)\\ &\quad=\bigg(\underline{\theta}^1\underline{b}^1,\ldots,\underline{\theta}^n\underline{b}^n,t_1+\log \frac{r_{\underline{b}^1}^{\underline{\theta}^1}}{r_{\underline{b}^n}^{\underline{\theta}^n}},\ldots,t_{n-1}+\log \frac{r_{\underline{b}^{n-1}}^{\underline{\theta}^{n-1}}}{r_{\underline{b}^n}^{\underline{\theta}^n}},v_1+v_{\underline{b}^1}^{\underline{\theta}^1},\ldots,v_n+v_{\underline{b}^n}^{\underline{\theta}^n}\bigg). \end{align*} $$

These are called renormalization operators. They will appear in the statement of the scale recurrence lemma. For $r>0$ , we also define the set

$$ \begin{align*}J_r=\{(t_1,\ldots,t_{n-1})\in\mathbb{R}^{n-1}: |t_j|\leq r, 1\leq j\leq n-1\}\times \mathbb{T}^n,\end{align*} $$

and denote by $\nu $ the Haar measure on J giving measure $(2\pi )^n$ to the set $J_{1/2}$ .

2.6 Scale recurrence lemma

In this subsection, we state one of the principal results in the paper. This is a multidimensional conformal version of the scale recurrence lemma of Moreira and Yoccoz [Reference Moreira and Yoccoz7]. The proof is technical and will be left for the end of the paper.

Theorem A. (Scale recurrence lemma)

Let $K_1$ , $K_2,\ldots ,K_{n}$ be $C^m$ conformal regular Cantor sets with $m\geq 2$ . Suppose they are not essentially affine and not essentially real. Denote by $d_j$ the Hausdorff dimension of $K_j$ , $1\leq j\leq n$ . If $r,c_0$ are conveniently large, there exist $c_1,c_2,c_3,\rho _0>0$ with the following properties: given $0<\rho <\rho _0$ , and a family $F(\underline {a}^1,\ldots ,\underline {a}^n)$ of subsets of $J_r$ , $(\underline {a}^1,\ldots ,\underline {a}^n)\in \Sigma _1(c_0,\rho )\times \cdots \times \Sigma _n(c_0,\rho )$ , such that

$$ \begin{align*}\nu(J_r\setminus F(\underline{a}^1,\ldots,\underline{a}^n))\leq c_1 \quad\text{for all } (\underline{a}^1,\ldots,\underline{a}^n),\end{align*} $$

there is another family $F^*(\underline {a}^1,\ldots ,\underline {a}^n)$ of subsets of $J_r$ satisfying the following properties.

1. (i) For any $(\underline {a}^1,\ldots ,\underline {a}^n)$ , $F^*(\underline {a}^1,\ldots ,\underline {a}^n)$ is contained in the $c_2 \rho $ -neighborhood of $F(\underline {a}^1,\ldots ,\underline {a}^n)$ .

2. (ii) Let $(\underline {a}^1,\ldots ,\underline {a}^n)\in \Sigma _1(c_0,\rho )\times \cdots \times \Sigma _n(c_0,\rho )$ , $(t,v)\in F^*(\underline {a}^1,\ldots ,\underline {a}^n)$ ; there exist at least $c_3\rho ^{-(d_1+d_2+\cdots +d_n)}$ tuples $(\underline {b}^1,\ldots ,\underline {b}^n)\in \Sigma _1(c_0,\rho )\times \cdots \times \Sigma _n(c_0,\rho )$ (with $\underline {b}^1,\ldots ,\underline {b}^n$ starting with the last letter of $\underline {a}^1,\ldots ,\underline {a}^n$ ) such that, if $\underline {\theta }^i \in \Sigma _i^{-}$ ends with $\underline {a}^i$ , $i=1,\ldots,n$ , and

   $$ \begin{align*}T_{\underline{b}^1,\ldots,\underline{b}^n}(\underline{\theta}^1,\ldots,\underline{\theta}^n,t,v)=(\underline{\theta}^1\underline{b}^1,\ldots,\underline{\theta}^n\underline{b}^n,\tilde{t},\tilde{v}),\end{align*} $$
   the $\rho $ -neighborhood of $(\tilde {t},\tilde {v})\in J$ is contained in $F^*(\underline {b}^1,\ldots ,\underline {b}^n)$ .

3. (iii) $\nu (F^*(\underline {a}^1,\ldots ,\underline {a}^n))\geq \nu (J_r)/2$ for at least half of the $(\underline {a}^1,\ldots ,\underline {a}^n)\in \Sigma _1(c_0,\rho )\times \cdots \times \Sigma _n(c_0,\rho )$ .

3.1 Discrete Marstrand property

In this section, we present and prove the discrete Marstrand property. We first state two linear algebra results that we will need.

Lemma 3.3. Let $E_1,E_2 \subset \mathbb {R}^n$ be linear subspaces such that $E_1+E_2=\mathbb {R}^n$ . Denote by $\theta $ the angle between $E_1$ and $E_2$ . Define $I=E_1\cap E_2$ , $L_1=I^{\perp }\cap E_1$ , and $L_2=I^{\perp }\cap E_2$ . For $x=l_1+v+l_2$ , with $l_1\in L_1$ , $l_2\in L_2$ , $v\in I$ , we have

$$ \begin{align*}|l_1|\leq \frac{\mathrm{dist}(x,E_2)}{\sin \theta}\quad \text{and}\quad |l_2|\leq \frac{\mathrm{dist}(x,E_1)}{\sin \theta}.\end{align*} $$

The next proposition is the main tool that will allow us to obtain the discrete Marstrand property. Given an $\mathbb {R}$ -linear map $B: \mathbb {C}^n\to \mathbb {R}^l$ , we will say it satisfies the transversality condition if for any set $A\subset \{1,\ldots ,n\}$ , we have $\textrm {dim}(Im(B|_{\{z_j=0:j\notin A\}}))=\min \{l,2\cdot \# A\};$ in all this subsection, B will denote one such map.

Proposition 3.1. Let $r>0$ and $B:\mathbb {C}^n\to \mathbb {R}^l$ a linear map satisfying the transversality condition. There exists a constant C, depending only on r and B, such that for any pair of subsets $Q_1,Q_2 \subset \mathbb {C}^n$ , we have

$$ \begin{align*}\nu(\{s\in J_r: B\circ A(s)(Q_1) \cap B\circ A(s)(Q_2)\neq \emptyset\})\leq C \bigg(\frac{\max \{\mathrm{diam}(Q_1),\mathrm{diam}(Q_2)\}}{\mathrm{dist}(Q_1,Q_2)}\bigg)^l\!,\end{align*} $$

where $\mathrm {dist}(Q_1,Q_2)$ denotes the distance between $Q_1$ and $Q_2$ .

Proof. Throughout the proof, we will use the notation $P=O(Q)$ , meaning that there is a constant $\tilde {C}$ , depending only on r and B, such that $P\leq \tilde {C} \cdot Q$ .

Given a subset $A\subset \{1,2,\ldots ,n\}$ , we consider the subspace $\mathbb {C}^A=\{(z_1,\ldots ,z_n)\in \mathbb {C}^n: z_j=0\,\text { for all } j\notin A\}.$ By the transversality condition, we can choose $\theta>0$ , only depending on B, such that the angle between $\mathrm {ker}(B)$ and $\mathbb {C}^A$ is bigger than $\theta $ for any non-empty subset A.

Denote $\max \{\mathrm {diam}(Q_1),\mathrm {diam}(Q_2)\}$ by $\rho $ . Fix $c_1,c_2 \in \mathbb {C}^n$ such that

$$ \begin{align*}\mathrm{dist}(c_j,Q_j)< \rho,\quad \mathrm{dist}(Q_1,Q_2)/2\leq |c_2-c_1|\leq 2\, \mathrm{dist}(Q_1,Q_2)\end{align*} $$

and $c_2-c_1$ has all its coordinates in $\mathbb {C}^n$ different from zero. Suppose that $s\in J_r$ is such that $BA(s)(Q_1) \cap BA(s)(Q_2)\neq \emptyset .$ Then there are $\tilde {c}_1\in Q_1$ , $\tilde {c}_2\in Q_2$ verifying $B\circ A(s)(\tilde {c}_1)=B\circ A(s)(\tilde {c}_2)$ . We conclude that

$$ \begin{align*}|B\circ A(s)(c_2-c_1)|=O(\rho).\end{align*} $$

Define $x=({c_2-c_1})/{|c_2-c_1|}$ , and hence $|B\circ A(s)(x)|=O({\rho }/{\mathrm {dist}(Q_1,Q_2)}).$ By the first linear algebra lemma, we get that $\mathrm {dist}(A(s)(x), \mathrm {ker}(B))=O({\rho }/{\mathrm {dist}(Q_1,Q_2)}).$ Up until now, we have proven that there is a constant $C_1$ , depending only on r and B, such that

$$ \begin{align*}\mathrm{dist}(A(s)(x), \mathrm{ker}(B))\leq C_1 \cdot \frac{\rho}{\mathrm{dist}(Q_1,Q_2)}.\end{align*} $$

Notice that if ${\rho }/{\mathrm {dist}(Q_1,Q_2)} \geq ({e^{-r} \sin \theta })/{4 C_1},$ then the proposition follows taking $C =(2r)^{n-1} (2\pi )^n ({4C_1}/{e^{-r} \sin \theta })^{l}$ . This is thanks to the fact that

$$ \begin{align*}\nu(\{s\in J_r: B\circ A(s)(Q_1) \cap B\circ A(s)(Q_2)\neq \emptyset\})\leq (2r)^{n-1} (2\pi)^n.\end{align*} $$

For the rest of the proof, we suppose ${\rho }/{\mathrm {dist}(Q_1,Q_2)} < ({e^{-r} \sin \theta })/{4 C_1}$ . Define $a=({e^{-r} \sin \theta })/{4n}$ and write $x\in \mathbb {C}^n$ as $x=(e^{\chi _1+i\phi _1},\ldots ,e^{\chi _n+i\phi _n}).$ Consider the set

$$ \begin{align*}A=\{j\in \{1,\ldots,n\}:e^{\chi_j}\geq a e^{-r}\},\end{align*} $$

and the subspace $\mathbb {C}^A$ ; we will see that $\textrm {dim}_{\mathbb {R}}\mathbb {C}^A\geq l$ . Let $u\in \mathbb {C}^A$ be the orthogonal projection of $A(s)(x)$ in $\mathbb {C}^A$ . By the definition of $\mathbb {C}^A$ , we have $|A(s)x-u|< na$ . Given that $\|A(s)\|\geq e^{-r}$ (remember that $s\in J_r$ ) and $na<e^{-r}/2$ , we have

$$ \begin{align*}|u|\geq |A(s)(x)|-|A(s)x-u|> e^{-r}-n\cdot a > \tfrac{1}{2}e^{-r}.\end{align*} $$

If $\textrm {dim}_{\mathbb {R}} \mathbb {C}^A<l$ , the transversality implies $\mathbb {C}^A \cap \mathrm {ker}(B)=\{0\}$ and then, by the choice of $\theta $ , we would get $\mathrm {dist}(u,\mathrm {ker}(B))\geq |u|\sin \theta \geq ({e^{-r}\sin \theta })/{2}.$ However, this is not possible since

$$ \begin{align*} \mathrm{dist}(u,\mathrm{ker}(B))&\leq |u-A(s)(x)|+\mathrm{dist}(A(s)(x),\mathrm{ker}(B))\\ &\leq na+ C_1 \cdot \frac{\rho}{\mathrm{dist}(Q_1,Q_2)} < \frac{e^{-r}\sin \theta}{2}. \end{align*} $$

Given that $\textrm {dim}_{\mathbb {R}}\mathbb {C}^A\geq l$ , the transversality condition implies $\mathrm {ker}(B)+\mathbb {C}^A=\mathbb {C}^n$ . Let $L=\mathbb {C}^A\cap (\mathrm {ker}(B)\cap \mathbb {C}^A)^{\perp }$ . Define the $\mathbb {R}$ -linear function $\hat {x}:\mathbb {C}^n\to \mathbb {C}^n$ given by $\hat {x}(z_1,\ldots ,z_n)=(e^{\chi _1+i\phi _1}\cdot z_1,\ldots ,e^{\chi _n+i\phi _n}\cdot z_n).$ Notice that $A(s)(x)=\hat {x}([s])$ , where $[s]=(e^{t_1+iv_1},\ldots ,e^{t_{n-1}+iv_{n-1}},e^{iv_n}).$ Write $\hat {x}([s])=b_1+b_2$ , where $b_1\in L$ , $b_2\in \mathrm {ker}(B)$ . The second lemma in linear algebra implies

$$ \begin{align*}|b_1|\leq \frac{\mathrm{dist}(b_1,\mathrm{ker}(B))}{\sin \theta}=\frac{\mathrm{dist}(\hat{x}([s]),\mathrm{ker}(B))}{\sin \theta}=O\bigg(\frac{\rho}{\mathrm{dist}(Q_1,Q_2)}\bigg).\end{align*} $$

Given that $b_1\in \mathbb {C}^A$ , we get that $|\hat {x}^{-1}(b_1)|=O({\rho }/{\mathrm {dist}(Q_1,Q_2)})$ . Therefore, $[s]=\hat {x}^{-1}(b_1)+ \hat {x}^{-1}(b_2)$ implies

$$ \begin{align*}\mathrm{dist}([s],\hat{x}^{-1}(\mathrm{ker}(B)))=O\bigg(\frac{\rho}{\mathrm{dist}(Q_1,Q_2)}\bigg).\end{align*} $$

This last inequality tells us that the vector $[s]$ is close to a $2n-l$ subspace. Moreover, the last coordinate of this vector has modulus 1. This two properties will allow us to obtain the desired estimate.

Consider the set

$$ \begin{align*}H=\{(z_1,\ldots,z_n)\in \hat{x}^{-1}( \mathrm{ker}(B)):\,|z_n|=1,\,|z_j|\in [e^{-2r},e^{2r}],\,j=1,\ldots,n-1\}.\end{align*} $$

We have proven that there is a constant $C_2>0$ , depending only on B and r, such that $\mathrm {dist}([s],\hat {x}^{-1}(\mathrm {ker}(B))) \leq C_2 \cdot {\rho }/{\mathrm {dist}(Q_1,Q_2)}.$ Thus, there is a constant $C_3>0$ , depending only on B and r, such that

$$ \begin{align*}\mathrm{dist}([s],H) \leq C_3 \cdot \frac{\rho}{\mathrm{dist}(Q_1,Q_2)}.\end{align*} $$

In fact, let $u=(u_1,\ldots ,u_n)\in \hat {x}^{-1}(\mathrm {ker}(B))$ such that $|u-[s]|\leq C_2\cdot {\rho }/{\mathrm {dist}(Q_1,Q_2)}$ . We have

$$ \begin{align*}1-C_2 \cdot \frac{\rho}{\mathrm{dist}(Q_1,Q_2)}\leq |u_n| \leq 1+C_2 \cdot \frac{\rho}{\mathrm{dist}(Q_1,Q_2)},\end{align*} $$

and

$$ \begin{align*}e^{-r}-C_2 \cdot \frac{\rho}{\mathrm{dist}(Q_1,Q_2)} \leq |u_j|\leq e^r+C_2 \cdot \frac{\rho}{\mathrm{dist}(Q_1,Q_2)}\end{align*} $$

for $j=1,\ldots ,n-1$ . Therefore, if ${\rho }/{\mathrm {dist}(Q_1,Q_2)}$ is small enough, one has that $u/|u_n|$ is in H and $|[s]-(u/|u_n|)|=O({\rho }/{\mathrm {dist}(Q_1,Q_2)})$ . If ${\rho }/{\mathrm {dist}(Q_1,Q_2)}$ is big, the proposition follows choosing C properly, as it was done before when we considered the case ${\rho }/{\mathrm {dist}(Q_1,Q_2)} \geq ({e^{-r} \sin \theta })/{4 C_1}$ .

Define the function

$$ \begin{align*}\varphi:\bigg\{(z_1,\ldots,z_n) \in \mathbb{S}^{2n-1} \cap \hat{x}^{-1}( \mathrm{ker}(B)): \,\frac{|z_j|}{|z_n|}\in [e^{-2r},e^{2r}],\,j=1,\ldots,n-1\bigg\} \to H,\end{align*} $$

given by $\varphi (z_1,\ldots ,z_n)=(z_1/|z_n|,\ldots ,z_n/|z_n|).$ Notice that $\varphi $ is surjective and smooth. Moreover, one has that $\|D\varphi \|$ is bounded by a constant depending only on r and n. Since the domain of $\varphi $ is contained in a $(2n-l-1)$ -dimensional unit sphere inside $\hat {x}^{-1}(\mathrm {ker}(B))$ , there exist $w_1,\ldots ,w_p\in H$ such that H is covered by the balls $B_{{\rho }/{\mathrm {dist}(Q_1,Q_2)}}(w_j)$ , $j=1,\ldots ,p$ , and

$$ \begin{align*}p=O\bigg(\bigg(\frac{\rho}{\mathrm{dist}(Q_1,Q_2)}\bigg)^{-(2n-l-1)}\bigg).\end{align*} $$

We conclude that $[s]\in \bigcup _{j=1}^{p} B_{C_4\cdot {\rho }/{\mathrm {dist}(Q_1,Q_2)}}(w_j),$ for a constant $C_4>0$ depending only on r and B.

Writing $w_j$ as $w_j=(w_{j,1},\ldots ,w_{j,n}),$ we obtain that for some $1\leq j\leq p$ ,

$$ \begin{align*}|e^{iv_n}-w_{j,n}|\leq C_4\cdot \frac{\rho}{\mathrm{dist}(Q_1,Q_2)},\end{align*} $$

$$ \begin{align*}|e^{t_q+iv_q}-w_{j,q}|\leq C_4\cdot \frac{\rho}{\mathrm{dist}(Q_1,Q_2)},\end{align*} $$

$q=1,\ldots ,n-1$ . Notice that $|w_{j,q}|$ is bounded below by $e^{-2r}$ , and hence

$$ \begin{align*}|t_q-\log |w_{j,q}|| \leq C_5\cdot \frac{\rho}{\mathrm{dist}(Q_1,Q_2)},\quad \|v_q-\mathrm{arg}(w_{j,q})\|\leq C_5\cdot \frac{\rho}{\mathrm{dist}(Q_1,Q_2)},\end{align*} $$

$q=1,\ldots ,n$ , where $C_5$ is a constant depending only on r and B, and $\text {arg}(w_{j,q})\in \mathbb {T}$ is the argument of $w_{j,q}$ . This implies that the set $\{s\in J_r: B\circ A(s)(Q_1) \cap B\circ A(s)(Q_2)\neq \emptyset \}$ is contained in the union of p sets, each one with a $\nu $ -volume of order $O ( ( {\rho }/{\mathrm {dist}(Q_1,Q_2)})^{2n-1} )$ . Finally, using the order of p, we conclude that

$$ \begin{align*}\nu(\{s\in J_r: B\circ A(s)(Q_1) \cap B\circ A(s)(Q_2)\neq \emptyset\})= O \bigg(\bigg(\frac{\rho}{\mathrm{dist}(Q_1,Q_2)}\bigg)^{(2n-1)-(2n-l-1)}\bigg),\end{align*} $$

as we wanted.

Proposition 3.1 implies that there is a constant $C>0$ such that for any $\underline {\theta }^1,\ldots ,\underline {\theta }^n$ and $Q_1,Q_2\in R( \rho )$ , we have

$$ \begin{align*}\nu(\{s\in J_r: B\circ \pi_{\underline{\theta}^1,\ldots,\underline{\theta}^n,s}(Q_1) \cap B\circ \pi_{\underline{\theta}^1,\ldots,\underline{\theta}^n,s}(Q_2)\neq \emptyset\})\leq C \bigg(\frac{\rho}{\mathrm{ dist}(Q_1,Q_2)}\bigg)^l.\end{align*} $$

The constant $C>0$ depends on B, r, $c_0$ , and the Cantor sets $K_1,\ldots ,K_n$ , but it is independent of $\rho $ . For the next proposition, we will also need the following fact: if p is big enough, then for any $Q_1\in R(\rho )$ and $a\in \mathbb {Z}$ , we have

$$ \begin{align*}\# \{Q_2\in R(\rho):p^{-a}\leq \mathrm{dist}(Q_1,Q_2)< p^{-a+1}\}=O((p^{-a})^{d_1+\cdots+d_n}\rho^{-(d_1+\cdots+d_n)}),\end{align*} $$

where $d_j=HD(K_j)$ , $j=1,\ldots ,n$ . For a proof, see [Reference Zamudio13, Lemma 1.2.3].

Proposition 3.2. Assume $d_1+\cdots +d_n< l$ . Let

$$ \begin{align*}&N_{\rho}(\underline{\theta}^1,\ldots,\underline{\theta}^n,s)\\&\quad=\# \{(Q_1,Q_2)\in R(\rho)^2: B\circ \pi_{\underline{\theta}^1,\ldots,\underline{\theta}^n,s}(Q_1) \cap B\circ \pi_{\underline{\theta}^1,\ldots,\underline{\theta}^n,s}(Q_2)\neq \emptyset \}.\end{align*} $$

Then for any $\underline {\theta }^1,\ldots ,\underline {\theta }^n$ , we have $\int _{J_r}N_{\rho }(\underline {\theta }^1,\ldots ,\underline {\theta }^n,s)ds=O(\rho ^{-(d_1+\cdots +d_j)}),$ and the constant in the O notation is independent of $\underline {\theta }^1,\ldots ,\underline {\theta }^n$ .

Proof. Since the Cantor sets $K_1,\ldots ,K_n$ have bounded geometries, then there is a constant $C_1$ , independent of $\rho $ , such that $\mathrm {dist}(Q_1,Q_2)\geq C_1 \rho $ for any $Q_1,Q_2\in R(\rho )$ , $Q_1\cap Q_2=\emptyset $ . Let $k_0\in \mathbb {Z}$ such that $p^{-k_0}\leq C_1 \rho < p^{-k_0+1}$ . Using the previous lemma, we have

$$ \begin{align*} &\int_{J_r} N_{\rho}(\underline{\theta}^1,\ldots,\underline{\theta}^n,s)\,ds\\ &\quad= \sum_{Q_1,Q_2 \in R(\rho)} \nu(\{s\in J_r: B\circ \pi_{\underline{\theta}^1,\ldots,\underline{\theta}^n,s}(Q_1) \cap B\circ \pi_{\underline{\theta}^1,\ldots,\underline{\theta}^n,s}(Q_2)\neq \emptyset \})\\ &\quad=\sum_{Q_1 \in R(\rho)} \sum_{k=-\infty}^{k_0} \sum_{\mathrm{dist}(Q_1,Q_2)\in [p^{-k},p^{-k+1})} O(\rho^l/[\mathrm{dist}(Q_1,Q_2)]^l)\\ &\qquad+ \sum_{Q_1 \in R(\rho)} \sum_{Q_2 \cap Q_1\neq \emptyset} (2r)^{n-1}. \end{align*} $$

Clearly, $\sum _{Q_1 \in R(\rho )} \sum _{Q_2 \cap Q_1\neq \emptyset } (2r)^{n-1}=O(\# R(\rho ))=O(\rho ^{-(d_1+\cdots +d_n)}).$ However,

$$ \begin{align*} &\sum_{Q_1 \in R(\rho)} \sum_{k=-\infty}^{k_0} \sum_{\mathrm{dist}(Q_1,Q_2)\in [p^{-k},p^{-k+1})} O(\rho^l/[\mathrm{dist}(Q_1,Q_2)]^l)\\ &\quad=\sum_{Q_1 \in R(\rho)} \sum_{k=-\infty}^{k_0} O((p^k)^{l-(d_1+\cdots+d_n)}\rho^{l-(d_1+\cdots+d_n)})\\ &\quad=\sum_{Q_1 \in R(\rho)} O((p^{k_0})^{l-(d_1+\cdots+d_n)}\rho^{l-(d_1+\cdots+d_n)})\sum_{k=-\infty}^{0} (p^{l-(d_1+\cdots+d_n)})^k\\ &\quad= O(\rho^{-(d_1+\cdots+d_n)}\rho^{-l+(d_1+\cdots+d_l)}\rho^{l-(d_1+\cdots+d_n)})=O(\rho^{-(d_1+\cdots+d_n)}).\\[-3pc] \end{align*} $$

Proof. For $Q_0\in F$ , define

$$ \begin{align*}n(Q_0)=\#\{Q\in F: B\circ \pi_{\underline{\theta}^1,\ldots,\underline{\theta}^n,s}(Q) \cap B\circ \pi_{\underline{\theta}^1,\ldots,\underline{\theta}^n,s}(Q_0)\neq \emptyset\}.\end{align*} $$

We have $\sum _{Q \in F} n(Q)\leq N_{\rho }(\underline {\theta }^1,\ldots ,\underline {\theta }^n,s) \leq a \rho ^{-(d_1+\cdots +d_n)}. $ Therefore, the set $T_0=\{Q\in F: n(Q)\leq 2a/b\}$ has at least $(1/2)\#F$ elements. Finally, it is clear that from $T_0$ , we can extract a subset T with at least $({1}/({2a/b}))\#T_0$ elements and such that $B\circ \pi _{\underline {\theta }^1,\ldots ,\underline {\theta }^n,s}(Q_1) \cap B\circ \pi _{\underline {\theta }^1,\ldots ,\underline {\theta }^n,s}(Q_2)= \emptyset $ for any $Q_1,Q_2 \in T$ . For this set, we have

$$ \begin{align*}\#T \geq \frac{1}{2a/b}\#T_0 \geq \frac{1}{2a/b}\frac{1}{2}b \rho^{-(d_1+\cdots+d_n)}=\frac{b^2}{4a}\rho^{-(d_1+\cdots+d_n)}.\\[-38pt]\end{align*} $$

Notice that since $\int _{J_r} N_{\rho }(\underline {\theta }^1,\ldots ,\underline {\theta }^n,s)\,ds=O(\rho ^{-(d_1+\cdots +d_m)})$ , then choosing a big enough, we can guarantee that the set

$$ \begin{align*}\{s\in J_r: N_{\rho}(\underline{\theta}^1,\ldots,\underline{\theta}^n,s)> a\rho^{-(d_1+\cdots+d_n)}\}\end{align*} $$

has measure as small as we want. Thus, for every $(\underline {\theta }^1,\ldots ,\underline {\theta }^n)$ , we have that most of the $s\in J_r$ verify the property of the last proposition, that is, for any family $F\subset R(\rho )$ with $\# F\geq b\rho ^{-(d_1+\cdots +d_n)}$ , there exist a positive proportion of F, $T\subset F$ with $\#T\geq (b^2/(4a))\rho ^{-(d_1+\cdots +d_n)}$ , such that elements of T project to $\mathbb {R}^l$ , in the direction of s, to disjoint sets: $B\circ \pi _{\underline {\theta }^1,\ldots ,\underline {\theta }^n,s}(Q_1) \cap B\circ \pi _{\underline {\theta }^1,\ldots ,\underline {\theta }^n,s}(Q_2)=\emptyset \ \text { for all } Q_1,Q_2 \in T$ . This is what we call the discrete Marstrand property.

The next lemma guarantees that the property of Proposition 3.3 still holds for small perturbations of $B\circ \pi _{\underline {\theta }^1,\ldots ,\underline {\theta }^n,s}$ ; it is inspired in the presentation given by Shmerkin [Reference Shmerkin12].

Lemma 3.4. Let $T\subset R(\rho )$ , $\phi $ a function defined on a neighborhood of $\bigcup _{Q\in T} Q$ into $\mathbb {R}^l$ , and $L,\tau>0$ real numbers. Suppose that for each $Q\in T$ , we have $B_{L^{-1}\rho }(c_Q)\subset \phi (Q) \subset B_{L\rho }(c_Q)$ for some $c_Q\in \mathbb {R}^l$ , and $\phi (Q_1)\cap \phi (Q_2)= \emptyset \ \text { for all } Q_1,Q_2\in T$ , $Q_1\neq Q_2$ . Then for any $\psi $ defined on a neighborhood of $\bigcup _{Q\in T} Q$ , such that the supremum norm $\|\phi - \psi \|<\tau \rho $ , there exists $T'\subset T$ such that

$$ \begin{align*}\#T'\geq [3L(L+\tau)]^{-l}\cdot \#T\end{align*} $$

and $\psi (Q_1)\cap \psi (Q_2)= \emptyset \ \text { for all } Q_1,Q_2\in T'$ , $Q_1\neq Q_2$ .

Proof. Here, $\|\psi -\phi \|<\tau \rho $ implies $\psi (Q)\subset B_{(L+\tau )\rho }(c_Q)$ . Use the Vitali covering lemma for the family $\{B_{(L+\tau )\rho }(c_Q):Q\in T\}$ . We get $T'\subset T$ such that $\{B_{(L+\tau )\rho }(c_Q):Q\in T'\}$ is a pairwise disjoint family and $\bigcup _{Q\in T} B_{(L+\tau )\rho }(c_Q)\subset \bigcup _{Q\in T'} B_{3(L+\tau )\rho }(c_Q).$ From this, we get

$$ \begin{align*} \# T' \cdot [3(L+\tau)\rho]^l w_l &\geq \mathrm{Vol}\bigg(\bigcup_{Q\in T} B_{(L+\tau)\rho}(c_Q)\bigg) \\ &\geq \mathrm{Vol}\bigg(\bigcup_{Q\in T} B_{L^{-1}\rho}(c_Q)\bigg) \\ &=\#T\cdot L^{-l}\rho^l w_l, \end{align*} $$

where $w_l$ is the volume of the l-dimensional unitary ball. Hence,

$$ \begin{align*}\#T'\geq [3L(L+\tau)]^{-l}\cdot \#T.\\[-35pt]\end{align*} $$

3.2 Proof of the dimension formula

In this subsection, we prove the desired dimension formula (Theorem B). Assume we have $K_1,\ldots ,K_n$ satisfying the hypothesis of the scale recurrence lemma. We start by using the discrete Marstrand property and the scale recurrence lemma to obtain, for each limit geometry $(\underline {\theta }^1,\ldots ,\underline {\theta }^n)$ , a set of ‘good’ directions to project. Fix $c_0,r>0$ big enough, and let $c_1,c_2,c_3,\rho _0$ be the constants given by the scale recurrence lemma. Suppose that h is a $C^{1}$ function defined from a neighborhood of $K_1\times \cdots \times K_n$ into $\mathbb {R}^l$ such that there exists a point $x_0\in K_1\times \cdots \times K_n$ , where $B=Dh(x_0)$ verifies the transversality hypotheses. By the results in §3.1, we can fix $a>0$ big enough such that

$$ \begin{align*}\nu(J_r\setminus \{s\in J_r: N_{\rho}(\underline{\theta}^1,\ldots,\underline{\theta}^n,s)\leq a\rho^{-(d_1+\cdots+d_n)}\})<c_1\end{align*} $$

for all $(\underline {\theta }^1,\ldots ,\underline {\theta }^n)$ . Define

$$ \begin{align*}F(\underline{\theta}^1,\ldots,\underline{\theta}^n)=\{s\in J_r: N_{\rho}(\underline{\theta}^1,\ldots,\underline{\theta}^n,s)\leq a\rho^{-(d_1+\cdots+d_n)}\},\end{align*} $$

and for $(\underline {a}^1,\ldots ,\underline {a}^n)\in \Sigma _{1}(c_0,\rho )\times \cdots \times \Sigma _{n}(c_0,\rho )$ ,

$$ \begin{align*}F(\underline{a}^1,\ldots,\underline{a}^n)=\bigcup_{\underline{\theta}^1,\ldots,\underline{\theta}^n} F(\underline{\theta}^1,\ldots,\underline{\theta}^n),\end{align*} $$

where the union is over all $\underline {\theta }^1,\ldots ,\underline {\theta }^n$ ending in $\underline {a}^1,\ldots ,\underline {a}^n$ , respectively. We clearly have $\nu (J_r\setminus F(\underline {a}^1,\ldots ,\underline {a}^n))<c_1$ , and thus we can apply the scale recurrence lemma (from now on, we assume $\rho <\rho _0$ ) to obtain sets $F^*(\underline {a}^1,\ldots ,\underline {a}^n)$ with the three properties in Theorem A (see §2.6).

Proof. Since h is Lipschitz in a neighborhood of $K_1\times \cdots \times K_n$ and $HD(K_1\times \cdots \times K_n)=d_1+\cdots +d_n$ , we have $HD(h(K_1\times \cdots \times K_n))\leq d_1+\cdots +d_n.$ Thus, we only need to show $HD(h(K_1\times \cdots \times K_n))\geq d_1+\cdots +d_n$ . Let $\eta>0$ arbitrary, we will prove that $HD(h(K_1\times \cdots \times K_n))\geq d_1+\cdots +d_n-\eta ,$ this will finish the proof of the theorem.

Since $\Sigma _1^-\times \cdots \times \Sigma _n^-\times J_r$ is compact and $\pi _{\underline {\theta }^1,\ldots ,\underline {\theta }^n,s}$ depends continuously in $(\underline {\theta }^1,\ldots ,\underline {\theta }^n,s)$ , then we can choose a constant $L>0$ (only depending on r, $K_1,\ldots,K_n$ , and $Dh(x_0)$ ) such that

(3) $$ \begin{align} B_{L^{-1}\rho}(c_Q)\subset Dh(x_0)\circ \pi_{\underline{\theta}^1,\ldots,\underline{\theta}^n,s}(Q) \subset B_{L\rho}(c_Q)\quad\text{for all } Q\in R(\rho), \end{align} $$

for some $c_Q\in \mathbb {R}^l$ , which depends on $\underline {\theta }^1,\ldots ,\underline {\theta }^n,s$ and Q.

Choose $\tau>0$ big enough such that

(4) $$ \begin{align} \|Dh(x_0)\circ \pi_{\tilde{\underline{\theta}}^1,\ldots,\tilde{\underline{\theta}}^n,\tilde{s}}-Dh(x_0)\circ \pi_{\underline{\theta}^1,\ldots,\underline{\theta}^n,s}\|\leq \tfrac{1}{3}\tau \rho \end{align} $$

for all $\underline {\theta }^j,\tilde {\underline {\theta }}^j\in \Sigma _j^-$ , $j=1,\ldots ,n$ , $s,\tilde {s}\in J_r$ with $|s-\tilde {s}|<c_2\rho $ , $\underline {\theta }^j\wedge \tilde {\underline {\theta }}^j\in \Sigma _j(c_0,\rho )$ , $j=1,\ldots ,n$ .

Until now, all the statements where $\rho $ appeared were true for any value small enough. For the rest of the proof, we are going to fix a particular value, which we call $\rho _1$ to distinguish it from the ‘variable’ $\rho $ . It is chosen such that $\rho _1^{\eta }\leq C_3[3L(L+\tau )]^{-l}{c_3^2}/{4a},$ where $C_3>0$ is a constant, independent of $\rho $ , which will be fixed later (equation (5)).

We can choose $\underline {a}_0^1,\ldots ,\underline {a}_0^n$ and $\underline {\theta }_0^1,\ldots ,\underline {\theta }_0^n$ ending in it, respectively, with the following properties.

(a) The element $s_0=(\log ({r_{\underline {a}_0^1}}/{r_{\underline {a}_0^n}}),\ldots ,\log ({r_{\underline {a}_0^{n-1}}}/{r_{\underline {a}_0^n}}),v_{\underline {a}_0^1},\ldots ,v_{\underline {a}_0^n})$ is in $J_r$ .

(b) $x_{0}\in G(\underline {a}_0^1)\times \cdots \times G(\underline {a}_0^n)$ .

(c) For any $\underline {b}^1,\ldots ,\underline {b}^n$ , consider $\tilde {s}=(\log ({r_{\underline {a}_0^1\underline {b}^1}}/{r_{\underline {a}_0^n\underline {b}^n}}),\ldots ,\log ({r_{\underline {a}_0^{n-1}\underline {b}^{n-1}}}/{r_{\underline {a}_0^n\underline {b}^n}}), v_{\underline {a}_0^1\underline {b}^1},\ldots ,v_{\underline {a}_0^n\underline {b}^n}).$ Then

$$ \begin{align*}\| Dh(c_{\underline{a}_0^1\underline{b}^1},\ldots,c_{\underline{a}_0^n\underline{b}^n})\circ \pi_{\underline{\theta}_0^1\underline{b}^1,\ldots,\underline{\theta}_0^n\underline{b}^n,\tilde{s}} - Dh(x_0) \circ \pi_{T_{\underline{b}^1,\ldots,\underline{b}^n}(\underline{\theta}_0^1,\ldots,\underline{\theta}_0^n,s)}\| \leq \tfrac{1}{3}\tau \rho_1.\end{align*} $$

This is achieved by choosing very long words for $\underline {a}_0^1,\ldots ,\underline {a}_0^n$ . If the number of symbols in $\underline {a}_0^j$ , for $1\leq j\leq n$ , increases, then $|x_0-(c_{\underline {a}_0^1\underline {b}^1},\ldots ,c_{\underline {a}_0^n\underline {b}^n})|$ goes to zero; the same happens for the distance between the last coordinate of $T_{\underline {b}^1,\ldots ,\underline {b}^n}(\underline {\theta }_0^1,\ldots ,\underline {\theta }_0^n,s)$ and $\tilde {s}$ . This a consequence of item (b) and the fact that $r_{\underline {a}_0^j}r^{\underline {\theta }_0^j}_{\underline {b}^j}/r_{\underline {a}_0^j\underline {b}^j} \to 1$ , $\|v_{\underline {a}_0^j\underline {b}^j}-(v_{\underline {a}_0^j}+v^{\underline {\theta }_0^j}_{\underline {b}^j})\|\to 0$ as the number of symbols in $\underline {a}_0^j$ goes to infinity.

(d) For any $\underline {b}^1,\ldots ,\underline {b}^n$ such that $\tilde {s}\in J_r$ , there is an affine function L such that

$$ \begin{align*}\|L\circ h \circ (f_{\underline{a}_0^1\underline{b}^1},\ldots,f_{\underline{a}_0^n\underline{b}^n})-Dh(c_{\underline{a}_0^1\underline{b}^1},\ldots,c_{\underline{a}_0^n\underline{b}^n})\circ \pi_{\underline{\theta}_0^1\underline{b}^1,\ldots,\underline{\theta}_0^n\underline{b}^n,\tilde{s}}\|\leq \tfrac{1}{3}\tau \rho_1.\end{align*} $$

This is a consequence of Lemma 3.1.

By hypothesis, there exists $(\underline {c}^1,\ldots ,\underline {c}^n)\in \Sigma _1^\textit{fin}\times \cdots \times \Sigma _n^\textit{fin}$ such that

$$ \begin{align*}T_{\underline{c}^1,\ldots,\underline{c}^n}(\underline{\theta}_0^1,\ldots,\underline{\theta}_0^n,s_0)=(\underline{\theta}_0^1\underline{c}^1,\ldots,\underline{\theta}_0^n\underline{c}^n,\tilde{s}_0)\end{align*} $$

and $\tilde {s}_0\in F^*(\tilde {\underline {a}}_0^1,\ldots ,\tilde {\underline {a}}_0^n)$ for some $(\tilde {\underline {a}}_0^1,\ldots ,\tilde {\underline {a}}_0^n)$ for which $(\underline {\theta }_0^1\underline {c}^1,\ldots ,\underline {\theta }_0^n\underline {c}^n)$ ends in it.

We will define inductively a set $N\subset \Sigma _1^\textit{fin}\times \cdots \times \Sigma _n^\textit{fin}\times \Sigma _1^-\times \cdots \times \Sigma _n^-\times J_r$ . Every $p=(\underline {a}^1,\ldots ,\underline {a}^n,\underline {\theta }^1,\ldots ,\underline {\theta }^n,s)\in N$ should verify:

1. (i) $s\in F^*(\tilde {\underline {a}}^1,\ldots ,\tilde {\underline {a}}^n)$ for some $(\tilde {\underline {a}}^1,\ldots ,\tilde {\underline {a}}^n)$ such that $(\underline {\theta }^1,\ldots ,\underline {\theta }^n)$ ends in $(\tilde {\underline {a}}^1,\ldots ,\tilde {\underline {a}}^n)$ ;

2. (ii) $(\underline {a}^1,\ldots ,\underline {a}^n,\underline {\theta }^1,\ldots ,\underline {\theta }^n,s)=(\underline {a}_0^1\underline {b}^1,\ldots ,\underline {a}_0^n\underline {b}^n,T_{\underline {b}^1,\ldots ,\underline {b}^n}(\underline {\theta }_0^1,\ldots ,\underline {\theta }_0^n,s_0))$ for some $(\underline {b}^1,\ldots ,\underline {b}^n)\in \Sigma _1^\textit{fin}\times \cdots \times \Sigma _n^\textit{fin}$ .

For $p=(\underline {a}^1,\ldots ,\underline {a}^n,\underline {\theta }^1,\ldots ,\underline {\theta }^n,s)\in N$ , we will define a set $T'(p)\subset R(\rho _1)$ verifying:

1. (iii) $\#T'(p)\geq C_3^{-1}\rho _1^{\eta -(d_1+\cdots +d_n)}$ ;

2. (iv) $h\circ (f_{\underline {a}^1},\ldots ,f_{\underline {a}^n})(Q_1)\cap h\circ (f_{\underline {a}^1},\ldots ,f_{\underline {a}^n})(Q_2)=\emptyset $ for all $Q_1,Q_2\in T'(p)$ , $Q_1\neq Q_2$ ;

3. (v) for all $(\underline {b}^1,\ldots ,\underline {b}^n)\in T'(p)$ , we have $T_{\underline {b}^1,\ldots ,\underline {b}^n}(\underline {\theta }^1,\ldots ,\underline {\theta }^n,s)=(\underline {\theta }^1\underline {b}^1,\ldots , \underline {\theta }^n\underline {b}^n,\tilde {s})$ and $\tilde {s}\in F^*(\underline {b}^1,\ldots ,\underline {b}^n)$ .

Elements of N are defined inductively, that is, every element already defined as $p\in N$ generates new elements, which we call the children of p and denote by $Ch(p)$ . Thus, N has the structure of a rooted tree. The root of the tree is $p_0=(\underline {a}_0^1\underline {c}^1,\ldots ,\underline {a}_0^n\underline {c}^n,\underline {\theta }_0^1\underline {c}^1,\ldots ,\underline {\theta }_0^n\underline {c}^n,\tilde {s}_0)$ , the set $T'(p_0)$ is defined as described below.

Given $p=(\underline {a}^1,\ldots ,\underline {a}^n,\underline {\theta }^1,\ldots ,\underline {\theta }^n,s)$ verifying items (i), (ii) (as $p_0$ does), define $T'(p)$ in the following way.

By item (i), we know that $s\in F^*(\tilde {\underline {a}}^1,\ldots ,\tilde {\underline {a}}^n)$ , and hence the scale recurrence lemma implies that there exists a set $F\subset R(\rho _1)$ with $\#F\geq c_3\rho _1^{-(d_1+\cdots +d_n)}$ and such that item (v) holds for F.

Since $F^*(\tilde {\underline {a}}^1,\ldots ,\tilde {\underline {a}}^n)\subset V_{c_2\rho _1}(F(\tilde {\underline {a}}^1,\ldots ,\tilde {\underline {a}}^n))$ , then there exist $s'\in F(\tilde {\underline {a}}^1,\ldots ,\tilde {\underline {a}}^n)$ with $|s-s'|\leq c_2\rho _1$ . By the definition of $F(\tilde {\underline {a}}^1,\ldots ,\tilde {\underline {a}}^n)$ , we have $N_{\rho _1}(\tilde {\underline {\theta }}^1,\ldots ,\tilde {\underline {\theta }}^n,s')\leq a\rho _1^{-(d_1+\cdots +d_n)}$ for some $(\tilde {\underline {\theta }}^1,\ldots ,\tilde {\underline {\theta }}^n)$ that ends in $(\tilde {\underline {a}}^1,\ldots ,\tilde {\underline {a}}^n)$ .

Using Proposition 3.3, we obtain a set $T\subset F$ such that

$$ \begin{align*}&Dh(x_0)\circ \pi_{\tilde{\underline{\theta}}^1,\ldots,\tilde{\underline{\theta}}^n,s'}(Q_1) \cap Dh(x_0)\circ \pi_{\tilde{\underline{\theta}}^1,\ldots,\tilde{\underline{\theta}}^n,s'}(Q_2)= \emptyset\\ &\qquad\text{for all } Q_1,Q_2\in T,\quad Q_1\neq Q_2,\end{align*} $$

and $\#T\geq (c_3^2/(4a))\rho _1^{-(d_1+\cdots +d_n)}$ .

We want to use Lemma 3.4 for $\phi =Dh(x_0)\circ \pi _{\tilde {\underline {\theta }}^1,\ldots ,\tilde {\underline {\theta }}^n,s'}$ , $\psi =L\circ h\circ (f_{\underline {a}^1},\ldots ,f_{\underline {a}^n})$ , where L is some affine function, and the set T. Note first that $\underline {\theta }^j\wedge \tilde {\underline {\theta }}^j\in \Sigma _j(c_0,\rho _1)$ , $1\leq j \leq n$ , since both $\underline {\theta }^j$ and $\tilde {\underline {\theta }}^j$ end in $\tilde {\underline {a}}^j$ . Equation (4) implies then

$$ \begin{align*}\|Dh(x_0)\circ \pi_{\tilde{\underline{\theta}}^1,\ldots,\tilde{\underline{\theta}}^n,s'}-Dh(x_0)\circ \pi_{\underline{\theta}^1,\ldots,\underline{\theta}^n,s}\|\leq \tfrac{1}{3}\tau \rho_1.\end{align*} $$

However, item (ii) together with items (c) and (d) imply

$$ \begin{align*}\|L\circ h\circ (f_{\underline{a}^1},\ldots,f_{\underline{a}^n})-Dh(x_0)\circ \pi_{\underline{\theta}^1,\ldots,\underline{\theta}^n,s}\|\leq \tfrac{2}{3}\tau \rho_1.\end{align*} $$

We conclude

$$ \begin{align*}\|L\circ h\circ (f_{\underline{a}^1},\ldots,f_{\underline{a}^n})-Dh(x_0)\circ \pi_{\tilde{\underline{\theta}}^1,\ldots,\tilde{\underline{\theta}}^n,s'}\|\leq \tau \rho_1,\end{align*} $$

which together with equation (3) shows that we can use Lemma 3.4. Hence, there is a subset $T'(p)\subset T\subset F$ such that

$$ \begin{align*} &h\circ (f_{\underline{a}^1},\ldots,f_{\underline{a}^n})(Q_1)\cap h\circ (f_{\underline{a}^1},\ldots,f_{\underline{a}^n})(Q_1) = \emptyset\\ &\qquad\text{for all } Q_1,Q_2 \in T'(p), Q_1\neq Q_2,\end{align*} $$

and

$$ \begin{align*}\#T'(p)\geq [3L(L+\tau)]^{-l}\cdot\frac{c_3^2}{4a}\rho_1^{-(d_1+\cdots+d_n)}\geq C_3^{-1}\rho_1^{\eta-(d_1+\cdots+d_n)}.\end{align*} $$

In the way we have defined $T'(p)$ , it clearly verifies items (iii), (iv), and (v).

Given $p=(\underline {a}^1,\ldots ,\underline {a}^n,\underline {\theta }^1,\ldots ,\underline {\theta }^n,s)\in N$ , the children of p are defined by

$$ \begin{align*}Ch(p)=\{(\underline{a}^1\underline{b}^1,\ldots,\underline{a}^n\underline{b}^n,T_{\underline{b}^1,\ldots,\underline{b}^n}(\underline{\theta}^1,\ldots,\underline{\theta}^n,s)): (\underline{b}^1,\ldots,\underline{b}^n)\in T'(p)\}.\end{align*} $$

The children of p clearly satisfy items (i) and (ii).

Now that we have defined N, we can finish the proof. For each non-negative integer k, consider the set $I(k)$ of elements $p\in N$ generated in the k-step of the inductive process. (They are children of elements generated in the $(k-1)$ -step, and the only element in the 0-step is $p_0$ .) For each $p=(\underline {a}^1,\ldots ,\underline {a}^n,\underline {\theta }^1,\ldots ,\underline {\theta }^n,s)\in N$ , define the set

$$ \begin{align*}G(p)=h(G(\underline{a}^1)\times\cdots\times G(\underline{a}^n)) \subset \mathbb{R}^l.\end{align*} $$

We clearly have

$$ \begin{align*}\bigcap_{k\geq 0}\bigcup_{p\in I(k)}G(p)\subset h(K_1\times\cdots\times K_n).\end{align*} $$

The desired result, $HD(h(K_1\times \cdots \times K_n))\geq d_1+\cdots +d_n-\eta $ , follows from Corollary 2.1 if we can prove that

$$ \begin{align*}\sum_{q\in Ch(p)} \bigg(\frac{\mathrm{diam}(G(q))}{\mathrm{diam} (G(p))}\bigg)^{d_1+\cdots+d_n-\eta}\geq 1,\end{align*} $$

and each set $G(p)$ contains a ball with radius proportional to its diameter. All other requirements in the corollary are obviously verified.

Given $p=(\underline {a}^1,\ldots ,\underline {a}^n,\underline {\theta }^1,\ldots ,\underline {\theta }^n,s)\in N$ , properties (i), (ii) and the observation in property (c) imply that $s\in J_r$ and $s(\underline {a}^1,\ldots ,\underline {a}^n):=(\log ({r_{\underline {a}^1}}/{r_{\underline {a}^n}}),\ldots ,\log ({r_{\underline {a}^{n-1}}}/{r_{\underline {a}^n}}), v_{\underline {a}^1},\ldots , v_{\underline {a}^n})$ is very close to s. Thus, we can assume $s(\underline {a}^1,\ldots ,\underline {a}^n)\in J_{2r}$ and then we can think of $G(\underline {a}^1)\times \cdots \times G(\underline {a}^n)$ as being a square; indeed we have $\mathrm { diam}(G(\underline {a}^j))\leq e^{4r} \mathrm {diam}(G(\underline {a}^m))$ for any $j, m$ . This together with the fact that $Dh(x_0)$ verify that the transversality hypotheses allow us to conclude that $h(G(\underline {a}^1)\times \cdots \times G(\underline {a}^n))$ contains a ball with radius proportional to its diameter. Moreover,

$$ \begin{align*} (C_3')^{-1} \mathrm{diam}(G(\underline{a}^1)\times\cdots\times G(\underline{a}^n)) &\leq \mathrm{diam}(h(G(\underline{a}^1)\times\cdots\times G(\underline{a}^n)))\\ &\leq C_3' \mathrm{diam}(G(\underline{a}^1)\times\cdots\times G(\underline{a}^n)) \end{align*} $$

for a constant $C_3'>0$ , depending only on r, h, and the Cantor sets. However, we can choose $C_4'>0$ , independent of $\rho $ , such that $\mathrm {diam}(G(\underline {a}\underline {b}))\geq C^{\prime }_4 \rho \cdot \mathrm {diam} (G(\underline {a}))$ for any $\underline {a}\in \Sigma _j^\textit{fin}$ and $\underline {b}\in \Sigma _j(c_0,\rho )$ , $0 \leq j\leq n$ . Therefore, we can choose $C_3>0$ that does not depend on $\rho _1$ , such that

(5) $$ \begin{align} \bigg(\frac{\mathrm{diam}(G(q))}{\mathrm{diam} (G(p))}\bigg)^{d_1+\cdots+d_n-\eta}\geq C_3 \rho_1^{d_1+\cdots+d_n-\eta} \end{align} $$

for any $q\in Ch(p)$ . Now that $C_3$ has been chosen, we get

$$ \begin{align*}\sum_{q\in Ch(p)} \bigg(\frac{\mathrm{diam}(G(q))}{\mathrm{diam} (G(p))}\bigg)^{d_1+\cdots+d_n-\eta}&\geq \sum_{q\in Ch(p)} C_3 \rho_1^{d_1+\cdots+d_n-\eta}\\&= C_3 \rho_1^{d_1+\cdots+d_n-\eta}\cdot \#T'(p)\geq 1.\\[-3.1pc]\end{align*} $$

Theorem B. (Dimension formula)

Let $K_1,\ldots ,K_n$ be $C^m$ , $m\geq 2$ , conformal regular Cantor sets generated by expanding maps $g_1,\ldots ,g_n$ , respectively. Suppose all of them are not essentially affine. Assume that there exist periodic points $p_j \in K_j$ , with period $n_j$ for $1\leq j\leq n$ , such that if we write $Dg_j^{n_j}(p_j)=\frac {1}{r_j} R_{-v_j},$ where $R_v$ is the rotation matrix by an angle $v\in \mathbb {T}$ , then

$$ \begin{align*} (\log r_1,0,\ldots,0&;v_1,0,\ldots,0),\\ &\vdots \\ (0,\ldots,\log r_{n-1}&;0,\ldots,v_{n-1},0),\\ (-\log r_n,\ldots,-\log r_n &;0,\ldots,0,v_n) \end{align*} $$

generate a dense subgroup of J. Let h be any $C^{1}$ function defined on a neighborhood of $K_1\times \cdots \times K_n$ into $\mathbb {R}^l$ such that there exists a point $x_0\in K_1\times \cdots \times K_n$ where $Dh(x_0)$ verifies the transversality hypotheses, then

$$ \begin{align*}HD(h(K_1\times\cdots\times K_n))=\min \{l,HD(K_1)+\cdots+HD(K_n)\}.\end{align*} $$

Proof. We first treat the case $HD(K_1)+\cdots +HD(K_n)<l$ . Notice that $K_1,\ldots ,K_n$ verify the hypotheses of the scale recurrence lemma; the existence of the periodic points $p_j$ imply that all of the Cantor sets are not essentially real. The desired result follows from the preceding theorem if we show that for any $(\underline {\theta }^1,\ldots ,\underline {\theta }^n,s)\in \Sigma _1^-\times \cdots \times \Sigma _n^-\times J_r$ , there exists $(\underline {c}^1,\ldots ,\underline {c}^n)\in \Sigma _1^\textit{fin}\times \cdots \times \Sigma _n^\textit{fin}$ such that $T_{\underline {c}^1,\ldots ,\underline {c}^n}(\underline {\theta }^1,\ldots ,\underline {\theta }^n,s)=(\underline {\theta }^1\underline {c}^1,\ldots ,\underline {\theta }^n\underline {c}^n,\tilde {s})$ and $\tilde {s}\in F^*(\tilde {\underline {a}}^1,\ldots ,\tilde {\underline {a}}^n)$ for some $(\tilde {\underline {a}}^1,\ldots ,\tilde {\underline {a}}^n)$ for which $(\underline {\theta }^1\underline {c}^1,\ldots ,\underline {\theta }^n\underline {c}^n)$ ends in it.

Let $\underline {a}_j \in \Sigma _j^\textit{fin}$ be the word of length $n_j$ such that the periodic point $p_j$ corresponds to the sequence $\underline {a}_j \underline {a}_j \underline {a}_j\cdots $ . For a finite sequence $\underline {a}\in \Sigma _j^\textit{fin}$ and $k\in \mathbb {Z}^+$ , we are going to use the notation $\underline {a}^k=\underbrace {\underline {a} \underline {a}\cdots \underline {a}}_{k-times}.$

Choosing $c_0$ big enough and assuming $\rho $ is small, we can find $k_j\in \mathbb {Z}^+$ such that $\tilde {\underline {a}}_j:=\underline {a}_j^{k_j} \in \Sigma _j(c_0,\rho ).$ Define $\underline {\theta }_j=\cdots \tilde {\underline {a}}_j\cdots \tilde {\underline {a}}_j\in \Sigma _j^-.$

By properties (ii) and (iii) of the scale recurrence lemma, we know that there are $s_0, s_1 \in J$ , $\underline {b}_j\in \Sigma _j(c_0,\rho )$ and $\underline {c}_j \in \Sigma ^\textit{fin}_j$ , $1\leq j \leq n$ , such that $T_{\underline {c}_1\underline {b}_1,\ldots ,\underline {c}_n\underline {b}_n}(\underline {\theta }_1,\ldots ,\underline {\theta }_n,s_0)=(\underline {\theta }_1\underline {c}_1\underline {b}_1,\ldots ,\underline {\theta }_n\underline {c}_n\underline {b}_n,s_1),$ and the $\rho $ -neighborhood of $s_1$ is contained in $F^*(\underline {b}_1,\ldots ,\underline {b}_n)$ .

Thanks to the continuity of the map $\underline {\theta }\to k^{\underline {\theta }}$ , we can choose positive integers $l_1,\ldots ,l_n$ , depending on $\rho $ , such that for any $m_j>l_j$ , any $\underline {\theta }^j \in \Sigma _j^-$ , $1\leq j\leq n$ , and x in the $\rho /2$ -neighborhood of $s_0$ , we have

$$ \begin{align*}T_{\underline{c}_1\underline{b}_1,\ldots,\underline{c}_n\underline{b}_n}(\underline{\theta}^1\tilde{\underline{a}}_1^{m_1},\ldots,\underline{\theta}^n\tilde{\underline{a}}_n^{m_n},x)=(\underline{\theta}^1\tilde{\underline{a}}_1^{m_1}\underline{c}_1\underline{b}_1,\ldots,\underline{\theta}^n\tilde{\underline{a}}_n^{m_n}\underline{c}_n\underline{b}_n,\tilde{x})\end{align*} $$

and $\tilde {x} \in F^*(\underline {b}_1,\ldots ,\underline {b}_n)$ . (We assume $\underline {\theta }^j$ , $1\leq j \leq n$ , ends with the letters in which $\underline {a}_j$ , $1\leq j \leq n$ , starts, otherwise we consider $\underline {d}^j\tilde {\underline {a}}_j^m$ instead of $\tilde {\underline {a}}_j^m$ for some $\underline {d}^j$ , and the proof follows in the same way.)

Now notice that $r^{\underline {\theta }^j}_{\tilde {\underline {a}}^{l_j+m_j}}=r^{\underline {\theta }^j}_{\tilde {\underline {a}}^{l_j}}\cdot r^{\underline {\theta }^j\tilde {\underline {a}}^{l_j}}_{\tilde {\underline {a}}^{m_j}},$ and if $l_j$ is big enough, we have

$$ \begin{align*}\Big|\log r^{\underline{\theta}^j\tilde{\underline{a}}_j^{l_j}}_{\tilde{\underline{a}}_j^{m_j}}-\log r^{\underline{\theta}_j}_{\tilde{\underline{a}}_j^{m_j}}\Big|<\frac{\rho}{8(2n-1)}\end{align*} $$

for any $m_j\in \mathbb {Z}^+$ . Similar formulas hold for $v^{\underline {\theta }^j\tilde {\underline {a}}_j^{l_j}}_{\tilde {\underline {a}}_j^{m_j}}$ .

For $m_j\in \mathbb {Z}^+$ , $1\leq j\leq n$ , consider $T_{\tilde {\underline {a}}_1^{l_1+m_1},\ldots ,\tilde {\underline {a}}_j^{l_j+m_j}}(\underline {\theta }^1,\ldots ,\underline {\theta }^n,s)=(\tilde {\underline {\theta }}^1,\ldots ,\tilde {\underline {\theta }}^n,\tilde {s}).$ We have

$$ \begin{align*} \tilde{s}&=s+\Big(\!\log r_{\tilde{\underline{a}}_1^{l_1+m_1}}^{\underline{\theta}^1}-\log r_{\tilde{\underline{a}}_n^{l_n+m_n}}^{\underline{\theta}^n},\ldots, \log r_{\tilde{\underline{a}}_{n-1}^{l_{n-1}+m_{n-1}}}^{\underline{\theta}^{n-1}}-\log r_{\tilde{\underline{a}}_n^{l_n+m_n}}^{\underline{\theta}^n};v_{\tilde{\underline{a}}_1^{l_1+m_1}}^{\underline{\theta}^1},\ldots,v_{\tilde{\underline{a}}_n^{l_n+m_n}}^{\underline{\theta}^n}\!\Big)\\ &=s+\Big(\!\log r_{\tilde{\underline{a}}_1^{l_1}}^{\underline{\theta}^1}-\log r_{\tilde{\underline{a}}_n^{l_n}}^{\underline{\theta}^n},\ldots, \log r_{\tilde{\underline{a}}_{n-1}^{l_{n-1}}}^{\underline{\theta}^{n-1}}-\log r_{\tilde{\underline{a}}_n^{l_n}}^{\underline{\theta}^n};v_{\tilde{\underline{a}}_1^{l_1}}^{\underline{\theta}^1},\ldots,v_{\tilde{\underline{a}}_n^{l_n}}^{\underline{\theta}^n}\!\Big)\\ &\quad+\Big(\!\log r_{\tilde{\underline{a}}_1^{m_1}}^{\underline{\theta}^1\tilde{\underline{a}}_1^{l_1}}-\log r_{\tilde{\underline{a}}_n^{m_n}}^{\underline{\theta}^n\tilde{\underline{a}}_n^{l_n}},\ldots, \log r_{\tilde{\underline{a}}_{n-1}^{m_{n-1}}}^{\underline{\theta}^{n-1}\tilde{\underline{a}}_{n-1}^{l_{n-1}}}-\log r_{\tilde{\underline{a}}_n^{m_n}}^{\underline{\theta}^n\tilde{\underline{a}}_n^{l_n}};v_{\tilde{\underline{a}}_1^{m_1}}^{\underline{\theta}^1\tilde{\underline{a}}_1^{l_1}},\ldots,v_{\tilde{\underline{a}}_n^{m_n}}^{\underline{\theta}^n\tilde{\underline{a}}_n^{l_n}}\!\Big). \end{align*} $$

Now notice that

$$ \begin{align*}\Big(\log r_{\tilde{\underline{a}}_1^{m_1}}^{\underline{\theta}^1\tilde{\underline{a}}_1^{l_1}}-\log r_{\tilde{\underline{a}}_n^{m_n}}^{\underline{\theta}^n\tilde{\underline{a}}_n^{l_n}},\ldots, \log r_{\tilde{\underline{a}}_{n-1}^{m_{n-1}}}^{\underline{\theta}^{n-1}\tilde{\underline{a}}_{n-1}^{l_{n-1}}}-\log r_{\tilde{\underline{a}}_n^{m_n}}^{\underline{\theta}^n\tilde{\underline{a}}_n^{l_n}};v_{\tilde{\underline{a}}_1^{m_1}}^{\underline{\theta}^1\tilde{\underline{a}}_1^{l_1}},\ldots,v_{\tilde{\underline{a}}_n^{m_n}}^{\underline{\theta}^n\tilde{\underline{a}}_n^{l_n}}\Big)\end{align*} $$

is in the $\rho /4$ neighborhood of

$$ \begin{align*}\Big(\log r_{\tilde{\underline{a}}_1^{m_1}}^{\underline{\theta}_1}-\log r_{\tilde{\underline{a}}_n^{m_n}}^{\underline{\theta}_n},\ldots, \log r_{\tilde{\underline{a}}_{n-1}^{m_{n-1}}}^{\underline{\theta}_{n-1}}-\log r_{\tilde{\underline{a}}_n^{m_n}}^{\underline{\theta}_n};v_{\tilde{\underline{a}}_1^{m_1}}^{\underline{\theta}_1},\ldots,v_{\tilde{\underline{a}}_n^{m_n}}^{\underline{\theta}_n}\Big).\end{align*} $$

However,

$$ \begin{align*} &\Big(\log r_{\tilde{\underline{a}}_1^{m_1}}^{\underline{\theta}_1}-\log r_{\tilde{\underline{a}}_n^{m_n}}^{\underline{\theta}_n},\ldots, \log r_{\tilde{\underline{a}}_{n-1}^{m_{n-1}}}^{\underline{\theta}_{n-1}}-\log r_{\tilde{\underline{a}}_n^{m_n}}^{\underline{\theta}_n};v_{\tilde{\underline{a}}_1^{m_1}}^{\underline{\theta}_1},\ldots,v_{\tilde{\underline{a}}_n^{m_n}}^{\underline{\theta}_n}\Big)\\ &\quad= m_1 \Big(\log r_{\tilde{\underline{a}}_1}^{\underline{\theta}_1},0,\ldots,0;v_{\tilde{\underline{a}}_1}^{\underline{\theta}_1},0,\ldots,0\Big)\\&\qquad+\cdots+ m_{n-1}\Big(0,\ldots,\log r_{\tilde{\underline{a}}_{n-1}}^{\underline{\theta}_{n-1}};0,\ldots,v_{\tilde{\underline{a}}_{n-1}}^{\underline{\theta}_{n-1}},0\Big)\\ &\qquad+m_n\Big(-\log r_{\tilde{\underline{a}}_{n}}^{\underline{\theta}_{n}},\ldots,-\log r_{\tilde{\underline{a}}_{n}}^{\underline{\theta}_{n}};0,\ldots,0,v_{\tilde{\underline{a}}_{n}}^{\underline{\theta}_{n}}\Big). \end{align*} $$

Moreover, $r_{\tilde {\underline {a}}_j}^{\underline {\theta }_j} R_{v_{\tilde {\underline {a}}_j}^{\underline {\theta }_j}}=[Dg^{k_jn_j}(p_j)]^{-1}=r_j^{k_j}R_{k_jv_j}.$ Therefore, the density hypothesis in the theorem, together with the next lemma, implies that there exist $m_1,\ldots ,m_n \in \mathbb {Z}^+$ such that

$$ \begin{align*}(\log r_{\tilde{\underline{a}}_1^{m_1}}^{\underline{\theta}_1}-\log r_{\tilde{\underline{a}}_n^{m_n}}^{\underline{\theta}_n},\ldots, \log r_{\tilde{\underline{a}}_{n-1}^{m_{n-1}}}^{\underline{\theta}_{n-1}}-\log r_{\tilde{\underline{a}}_n^{m_n}}^{\underline{\theta}_n};v_{\tilde{\underline{a}}_1^{m_1}}^{\underline{\theta}_1},\ldots,v_{\tilde{\underline{a}}_n^{m_n}}^{\underline{\theta}_n})\end{align*} $$

is in the $\rho /4$ -neighborhood of

$$ \begin{align*}s_0-s-(\log r_{\tilde{\underline{a}}_1^{l_1}}^{\underline{\theta}^1}-\log r_{\tilde{\underline{a}}_n^{l_n}}^{\underline{\theta}^n},\ldots, \log r_{\tilde{\underline{a}}_{n-1}^{l_{n-1}}}^{\underline{\theta}^{n-1}}-\log r_{\tilde{\underline{a}}_n^{l_n}}^{\underline{\theta}^n};v_{\tilde{\underline{a}}_1^{l_1}}^{\underline{\theta}^1},\ldots,v_{\tilde{\underline{a}}_n^{l_n}}^{\underline{\theta}^n}).\end{align*} $$

Hence, $\tilde {s}$ is in the $\rho /2$ -neighborhood of $s_0$ and from this, we get

$$ \begin{align*} T_{\tilde{\underline{a}}_1^{l_1+m_1}\underline{c}_1\underline{b}_1,\ldots,\tilde{\underline{a}}_n^{l_n+m_n}\underline{c}_n\underline{b}_n}(\underline{\theta}^1,\ldots,\underline{\theta}^n,s)&= T_{\underline{c}_1\underline{b}_1,\ldots,\underline{c}_n\underline{b}_n}(\underline{\theta}^1\tilde{\underline{a}}_1^{l_1+m_1},\ldots,\underline{\theta}^n\tilde{\underline{a}}_n^{l_n+m_n},\tilde{s})\\ &=(\underline{\theta}^1\tilde{\underline{a}}_1^{l_1+m_1}\underline{c}_1\underline{b}_1,\ldots,\underline{\theta}^n\tilde{\underline{a}}_n^{l_n+m_n}\underline{c}_n\underline{b}_n,\tilde{s}'), \end{align*} $$

and $\tilde {s}'\in F^*(\underline {b}_1,\ldots ,\underline {b}_n)$ , as we wanted.

If $HD(K_1)+\cdots +HD(K_n)\geq l$ , fix $\epsilon>0$ and find conformal regular Cantor sets $\tilde {K}_j\subset K_j$ , $1\leq j\leq n$ such that $l-\epsilon <HD(\tilde {K}_1)+\cdots +HD(\tilde {K}_n)<l$ , $p_j\in \tilde {K}_j$ , and the expanding map of $\tilde {K}_j$ is given by a power of $g_j$ , $1\leq j \leq n$ (see [Reference Moreira8, Lemma in p. 16]). We get

$$ \begin{align*} l\geq HD(h(K_1\times \cdots\times K_n))&\geq HD(h(\tilde{K}_1 \times\cdots\times \tilde{K}_n))\\ &=HD(\tilde{K}_1)+\cdots+HD(\tilde{K}_n)>l-\epsilon. \end{align*} $$

Since $\epsilon $ can be arbitrarily small, we obtain $HD(h(K\times \cdots \times K_n))=l$ as we wanted.

The following lemma was used in the previous theorem. It also implies that the hypothesis needed for the dimension formula is generic. Its proof is based on the well-known Kronecker theorem.

Lemma 3.5. Let $\unicode{x3bb} _j<0$ , $v_j\in \mathbb {T}$ , $1\leq j\leq n$ , and consider the set $E(\unicode{x3bb} _1,\ldots ,\unicode{x3bb} _n,v_1,\ldots, v_n)\subset \mathbb {R}^{n-1}\times \mathbb {T}^{n}$ given by the vectors

We have the following properties.

1. (a) If $E(\unicode{x3bb} _1,\ldots ,\unicode{x3bb} _n,v_1,\ldots ,v_n)$ generates a dense subgroup of $\mathbb {R}^{n-1}\times \mathbb {T}^{n}$ , then $E(k_1 \unicode{x3bb} _1,\ldots ,k_n \unicode{x3bb} _n,k_1 v_1,\ldots ,k_n v_n)$ also generates a dense subgroup for all $k_1,\ldots ,k_n \in \mathbb {Z}\setminus \{0\}$ .

2. (b) If $E(\unicode{x3bb} _1,\ldots ,\unicode{x3bb} _n,v_1,\ldots ,v_n)$ generates a dense subgroup of $\mathbb {R}^{n-1}\times \mathbb {T}^{n}$ , then it also generates a dense semigroup, that is, the set of linear combinations of vectors in $E(\unicode{x3bb} _1,\ldots ,\unicode{x3bb} _n,v_1,\ldots ,v_n)$ with coefficients in $\mathbb {N}$ is dense in $\mathbb {R}^{n-1}\times \mathbb {T}^{n}$ .

3. (c) The set of values $(\unicode{x3bb} _1,\ldots ,\unicode{x3bb} _n,v_1,\ldots ,v_n)\in \mathbb {R}_{<0}^{n}\times \mathbb {T}^n$ , for which the set $E(\unicode{x3bb} _1,\ldots ,\unicode{x3bb} _n,v_1,\ldots ,v_n)$ generates a dense subgroup, is a countable intersection of open and dense subsets.

4.1 General setting

We proceed as in [Reference Moreira and Yoccoz7, 6.1] using Fourier analysis in the group J instead of $\mathbb {R}$ . Let A be a set of indices, $\Lambda $ a finite set, and maps $\alpha :\Lambda \to A$ , $\omega :\Lambda \to A$ . Define $\Lambda _i=\alpha ^{-1}(i)$ , $\Lambda ^j=\omega ^{-1}(j)$ , $\Lambda _i^j=\Lambda _i\cap \Lambda ^j$ , $N_i=\#\Lambda _i$ , $N_i^j=\#\Lambda _i^j$ , $p_i^j=N_i^j/N_i$ . The numbers $(p_i^j)$ define a stochastic matrix, it has a probability vector $(p^i)$ verifying $\sum _{i \in A}p^i p_i^j=p^j, \sum _{i\in A}p^i=1.$ Set

$$ \begin{align*} p_{\unicode{x3bb}}^{\unicode{x3bb}'}=\begin{cases} 0 & \mbox{if } \omega(\unicode{x3bb}) \neq \alpha(\unicode{x3bb}'), \\ 1/N_{\omega(\unicode{x3bb})} & \mbox{if } \omega(\unicode{x3bb}) = \alpha(\unicode{x3bb}'), \end{cases} \end{align*} $$

and $p^{\unicode{x3bb} }=\frac {p^{\alpha (\unicode{x3bb} )}}{N_{\alpha (\unicode{x3bb} )}}.$ It is easily proved that $(p_{\unicode{x3bb} }^{\unicode{x3bb} '})$ is a stochastic matrix with probability vector $(p^{\unicode{x3bb} })$ . Let $J^*=\mathbb {R}^{n-1}\times \mathbb {Z}^n$ denote the Pontryagin dual of J. Elements $\xi =(\mu _1,\ldots ,\mu _{n-1},m_1,\ldots ,m_n)\in J^*$ are homomorphisms from J to $\mathbb {S}^1$ , given by

$$ \begin{align*}\xi(t_1,\ldots,t_{n-1},v_1,\ldots,v_n)=e^{(\sum_{j=1}^{n-1}t_j\mu_j+ \sum_{j=1}^{n}m_jv_j)i}.\end{align*} $$

Now suppose that for each $(\unicode{x3bb} ,\unicode{x3bb} ')\in \Lambda ^2$ , there is an element $a_{\unicode{x3bb} }^{\unicode{x3bb} '} \in J$ . Using this, we define for each $\xi \in J^*$ , a linear operator $T_{\xi }:\mathbb {C}^{\Lambda }\to \mathbb {C}^{\Lambda }$ given by $T_{\xi }((z_{\unicode{x3bb} })_{\unicode{x3bb} \in \Lambda })=(w_{\unicode{x3bb} })_{\unicode{x3bb} \in \Lambda }$ , where $w_\unicode{x3bb} =\sum _{\unicode{x3bb} '\in \Lambda } p_{\unicode{x3bb} }^{\unicode{x3bb} '}\xi (a_{\unicode{x3bb} }^{\unicode{x3bb} '})z_{\unicode{x3bb} '}.$

We endow the space $\mathbb {C}^{\Lambda }$ with the norm $\|(z_{\unicode{x3bb} })_{\unicode{x3bb} \in \Lambda }\|^2=\sum _{\unicode{x3bb} \in \Lambda }p^{\unicode{x3bb} }|z_{\unicode{x3bb} }|^2.$ In a similar way to [Reference Moreira and Yoccoz7], a short computation shows that $\| T_{\xi }\|\leq 1$ for all $\xi \in J^*$ .

Assume that we have a family $\{E(\unicode{x3bb} )\}_{\unicode{x3bb} \in \Lambda }$ of bounded measurable subsets of J, and consider the function

$$ \begin{align*}n_{\unicode{x3bb}}(x)=\frac{1}{N_{\omega(\unicode{x3bb})}}\cdot \#\{\unicode{x3bb}'\in \Lambda_{\omega(\unicode{x3bb})}: B_{\rho}(x+a_{\unicode{x3bb}}^{\unicode{x3bb}'})\subset E(\unicode{x3bb}')\}.\end{align*} $$

Let $0<\tau <1$ , and denote by $E^*(\unicode{x3bb} )$ the set $E^*(\unicode{x3bb} )=\{x\in J: n_{\unicode{x3bb} }(x)>\tau \}.$

Proposition 4.1. Suppose there exist $\Delta _0>0$ and $k_{0}\in (0,1)$ such that $\| T_{\xi }\|<k_0$ for all $\xi =(\mu _1,\ldots ,\mu _{n-1},m_1,\ldots ,m_n)$ , with $|\xi |=\max _j \{|\mu _j|,|m_j|\}\in [1,\Delta _0 \rho ^{-1}]$ . Then there exist $k_1\in (0,1)$ , $\epsilon>0$ , and $\tau \in (0,1)$ depending only on $\Delta _0$ , $k_0$ (and not on $\rho $ ) such that if $\nu (E(\unicode{x3bb} ))\leq \epsilon $ for all $\unicode{x3bb} \in \Lambda $ , then

$$ \begin{align*}\sum_{\unicode{x3bb} \in \Lambda}p^{\unicode{x3bb}}\nu(E^*(\unicode{x3bb}))\leq k_1 \sum_{\unicode{x3bb} \in \Lambda}p^{\unicode{x3bb}}\nu(E(\unicode{x3bb})).\end{align*} $$

Proof. Consider the functions $X_{\unicode{x3bb} }=1_{E(\unicode{x3bb} )},\ Y_{\unicode{x3bb} }(x)=\sum _{\unicode{x3bb} '\in \Lambda }p_{\unicode{x3bb} }^{\unicode{x3bb} '}X_{\unicode{x3bb} '}(x+a_{\unicode{x3bb} }^{\unicode{x3bb} '}),$

$$ \begin{align*} Z_{\unicode{x3bb}}(x)=\frac{1}{\nu(B_{\rho}(0))}\cdot \int_{B_{\rho}(0)}Y_{\unicode{x3bb}}(x-t)d\nu(t)=\frac{1}{\nu(B_{\rho}(0))} 1_{B_{\rho}(0)}\ast Y_{\unicode{x3bb}}(x). \end{align*} $$

Note that $Z_{\unicode{x3bb} }(x)\geq n_{\unicode{x3bb} }(x)$ , then $\|Z_{\unicode{x3bb} }\|_{L^2}^2\geq \tau ^2 \nu (E^*(\unicode{x3bb} ))$ , which implies

(6) $$ \begin{align} \sum_{\unicode{x3bb}\in\Lambda}p^{\unicode{x3bb}}\nu(E^*(\unicode{x3bb}))\leq \tau^{-2}\sum_{\unicode{x3bb}\in\Lambda}p^{\unicode{x3bb}}\|Z_{\unicode{x3bb}}\|_{L^2}^2. \end{align} $$

The Fourier transforms of $X_{\unicode{x3bb} }$ , $Y_{\unicode{x3bb} }$ , $Z_{\unicode{x3bb} }$ are

$$ \begin{align*} \hat{X}_{\unicode{x3bb}}(\xi)&=\int_{J}X_{\unicode{x3bb}}(x)\bar{\xi}(x)d\nu(x),\,\, \hat{Y}_{\unicode{x3bb}}(\xi)= \sum_{\unicode{x3bb}'\in \Lambda}p_{\unicode{x3bb}}^{\unicode{x3bb}'}\xi(a_{\unicode{x3bb}}^{\unicode{x3bb}'})\hat{X}_{\unicode{x3bb}'}(\xi),\\ \hat{Z}_{\unicode{x3bb}}(\xi)&=\frac{1}{\nu(B_{\rho}(0))} \hat{1}_{B_{\rho}(0)}\cdot \hat{Y}_{\unicode{x3bb}}(\xi)= \prod_{j=1}^{n-1}\frac{\sin(\mu_j\rho)}{\mu_j\rho} \prod_{j=1}^{n}\frac{\sin(m_j\rho)}{m_j\rho}\cdot \hat{Y}_{\unicode{x3bb}}(\xi), \end{align*} $$

where $\xi =(\mu _1,\ldots ,\mu _{n-1},m_1,\ldots ,m_n)$ . Hence, $|\hat {Z}_{\unicode{x3bb} }(\xi )|\leq |\hat {Y}_{\unicode{x3bb} }(\xi )|$ , and there exists $\tilde {k}_1\in (0,1)$ , depending only on $\Delta _0$ , such that $|\hat {Z}_{\unicode{x3bb} }(\xi )|\leq \tilde {k}_1|\hat {Y}_{\unicode{x3bb} }(\xi )|$ if $|\xi |>\Delta _0 \rho ^{-1}$ . We estimate $\sum _{\unicode{x3bb} \in \Lambda }p^{\unicode{x3bb} }|\hat {Z}_{\unicode{x3bb} }(\xi )|^2$ in various ways depending on $|\xi |$ .

If $|\xi |<1$ , then

$$ \begin{align*} |\hat{Z}_{\unicode{x3bb}}(\xi)|\leq |\hat{Y}_{\unicode{x3bb}}(\xi)| \leq \int_{J}|Y_{\unicode{x3bb}}(x)|d\nu(x) \leq \sum_{\unicode{x3bb}'\in \Lambda}p_{\unicode{x3bb}}^{\unicode{x3bb}'}\nu(E(\unicode{x3bb}')), \end{align*} $$

therefore,

$$ \begin{align*} \sum_{\unicode{x3bb} \in \Lambda}p^{\unicode{x3bb}}|\hat{Z}_{\unicode{x3bb}}(\xi)|^2 \leq \sum_{\unicode{x3bb}' \in \Lambda} \bigg( \sum_{\unicode{x3bb} \in \unicode{x3bb}}p^{\unicode{x3bb}}p_{\unicode{x3bb}}^{\unicode{x3bb}'}\bigg)\nu(E(\unicode{x3bb}'))^2 = \sum_{\unicode{x3bb}\in \Lambda}p^{\unicode{x3bb}} \nu(E(\unicode{x3bb}))^2. \end{align*} $$

If $1\leq |\xi | \leq \Delta _{0}\rho ^{-1}$ ,

$$ \begin{align*} \sum_{\unicode{x3bb} \in \Lambda}p^{\unicode{x3bb}}|\hat{Z}_{\unicode{x3bb}}(\xi)|^2\leq \sum_{\unicode{x3bb} \in \Lambda}p^{\unicode{x3bb}}|\hat{Y}_{\unicode{x3bb}}(\xi)|^2 \leq k_0^2 \sum_{\unicode{x3bb} \in \Lambda}p^{\unicode{x3bb}}|\hat{X}_{\unicode{x3bb}}(\xi)|^2, \end{align*} $$

note that we used the fact that $(\hat {Y}_{\unicode{x3bb} }(\xi ))_{\unicode{x3bb} \in \Lambda }=T_{\xi }((\hat {X}_{\unicode{x3bb} }(\xi ))_{\unicode{x3bb} \in \Lambda })$ .

If $|\xi |>\Delta _{0}\rho ^{-1}$ ,

$$ \begin{align*} \sum_{\unicode{x3bb} \in \Lambda}p^{\unicode{x3bb}}|\hat{Z}_{\unicode{x3bb}}(\xi)|^2\leq \tilde{k}_1^2 \sum_{\unicode{x3bb} \in \Lambda}p^{\unicode{x3bb}}|\hat{Y}_{\unicode{x3bb}}(\xi)|^2 \leq \tilde{k}_1^2 \sum_{\unicode{x3bb} \in \Lambda}p^{\unicode{x3bb}}|\hat{X}_{\unicode{x3bb}}(\xi)|^2. \end{align*} $$

Combining all three inequalities, we get

$$ \begin{align*} &\int_{J^*}\sum_{\unicode{x3bb} \in \Lambda}p^{\unicode{x3bb}}|\hat{Z}_{\unicode{x3bb}}(\xi)|^2d\hat{\nu}(\xi) \\ &\quad\leq \int_{|\xi|<1} \sum_{\unicode{x3bb}\in \Lambda}p^{\unicode{x3bb}} \nu(E(\unicode{x3bb}))^2 d\hat{\nu}(\xi)+ k_{0}^2\int_{1\leq |\xi|\leq \Delta_0\rho^{-1}} \sum_{\unicode{x3bb} \in \Lambda}p^{\unicode{x3bb}}|\hat{X}_{\unicode{x3bb}}(\xi)|^2 d\hat{\nu}(\xi)\\ &\qquad+ \tilde{k}_1^2\int_{|\xi|>\Delta_0\rho^{-1}} \sum_{\unicode{x3bb} \in \Lambda}p^{\unicode{x3bb}}|\hat{X}_{\unicode{x3bb}}(\xi)|^2 d\hat{\nu}(\xi). \end{align*} $$

Hence,

$$ \begin{align*} \sum_{\unicode{x3bb} \in \Lambda}p^{\unicode{x3bb}}\|\hat{Z}_{\unicode{x3bb}}\|_{L^2}^2 \leq \hat{\nu}(\{|\xi|<1\})\cdot \epsilon \sum_{\unicode{x3bb}\in \Lambda}p^{\unicode{x3bb}} \nu(E(\unicode{x3bb}))+\max\{k_0^2,\tilde{k}_1^2\} \sum_{\unicode{x3bb} \in \Lambda}p^{\unicode{x3bb}}\|\hat{X}_{\unicode{x3bb}}\|_{L^2}^2.\end{align*} $$

Using Plancherel theorem and the fact that $\nu (E(\unicode{x3bb} ))=\|X_{\unicode{x3bb} }\|_{L^2}^2$ , we get

$$ \begin{align*} \sum_{\unicode{x3bb} \in \Lambda}p^{\unicode{x3bb}}\|Z_{\unicode{x3bb}}\|_{L^2}^2 \leq [\hat{\nu}(\{|\xi|<1\})\cdot \epsilon +\max\{k_0^2,\tilde{k}_1^2\}]\sum_{\unicode{x3bb}\in \Lambda}p^{\unicode{x3bb}} \nu(E(\unicode{x3bb})).\end{align*} $$

This together with equation (6) imply

$$ \begin{align*} \sum_{\unicode{x3bb}\in\Lambda}p^{\unicode{x3bb}}\nu(E^*(\unicode{x3bb}))\leq \tau^{-2} [\hat{\nu}(\{|\xi|<1\})\cdot \epsilon +\max\{k_0^2,\tilde{k}_1^2\}]\sum_{\unicode{x3bb}\in \Lambda}p^{\unicode{x3bb}} \nu(E(\unicode{x3bb})). \end{align*} $$

Finally, we get the desired inequality setting

$$ \begin{align*} \tau=[\hat{\nu}(\{|\xi|<1\})\cdot \epsilon +\max\{k_0^2,\tilde{k}_1^2\}]^{1/3}, k_1=\tau^{-2} [\hat{\nu}(\{|\xi|<1\})\cdot \epsilon +\max\{k_0^2,\tilde{k}_1^2\}], \end{align*} $$

and taking $\epsilon $ small enough such that $\hat {\nu }(\{|\xi |<1\})\cdot \epsilon +\max \{k_0^2,\tilde {k}_1^2\}<1.$

Let $\Delta _1>0$ be any positive number and consider

$$ \begin{align*}\hat{n}_{\unicode{x3bb}}(x)=\frac{1}{\# \Lambda_{\omega(\unicode{x3bb})}}\cdot \#\{\unicode{x3bb}'\in \Lambda_{\omega(\unicode{x3bb})}:B_{\Delta_1 \rho}(x+a_{\unicode{x3bb}}^{\unicode{x3bb}'})\cap E(\unicode{x3bb}')\neq \emptyset \};\end{align*} $$

define $\hat {E}(\unicode{x3bb} )=\{x\in J: \hat {n}_{\unicode{x3bb} }(x)>\tau \}$ . For a set $E\subset J$ , denote by $V_{\delta }(E)$ the $\delta $ -neighborhood of E.

Corollary 4.1. Under the same hypothesis as Proposition 4.1, let $k_4>0$ such that $k_1<k_4<1$ ; if we choose $\Delta>0$ big enough and $\epsilon _1>0$ small enough such that

$$ \begin{align*} k_1\bigg(1+\frac{1+\Delta_1}{\Delta}\bigg)^{2n-1}<k_4,\,\,\, \epsilon_1 \bigg(1+\frac{1+\Delta_1}{\Delta}\bigg)^{2n-1} < \epsilon, \end{align*} $$

then $\nu (V_{\Delta \rho }(E(\unicode{x3bb} )))\leq \epsilon _1$ , for all $\unicode{x3bb} \in \Lambda $ , implies that

$$ \begin{align*}\sum_{\unicode{x3bb} \in \Lambda}p^{\unicode{x3bb}}\nu(V_{\Delta \rho}(\hat{E}(\unicode{x3bb})))\leq k_4 \sum_{\unicode{x3bb} \in \Lambda}p^{\unicode{x3bb}}\nu(V_{\Delta \rho}(E(\unicode{x3bb}))).\end{align*} $$

Proof. First observe that (see [Reference Lowther5])

$$ \begin{align*}\nu(V_{\rho+\Delta\rho+\Delta_1\rho}(E(\unicode{x3bb})))\leq \bigg(1+\frac{1+\Delta_1}{\Delta}\bigg)^{2n-1} \nu(V_{\Delta\rho}(E(\unicode{x3bb}))).\end{align*} $$

Now consider the family $A(\unicode{x3bb} )=V_{\rho +\Delta \rho +\Delta _1\rho }(E(\unicode{x3bb} ))$ ; by our choice of $\epsilon _1$ , we can apply Proposition 4.1 to $A(\unicode{x3bb} )$ . Notice also that if $x\in V_{\Delta \rho }(\hat {E}(\unicode{x3bb} ))$ , then there exist $y\in B_{\Delta \rho }(x)$ and $\tau \cdot \#\Lambda _{\omega (\unicode{x3bb} )}$ elements $\unicode{x3bb} '\in \Lambda _{\omega (\unicode{x3bb} )}$ such that $B_{\Delta _1 \rho }(y+a_{\unicode{x3bb} }^{\unicode{x3bb} '})\cap E(\unicode{x3bb} ')\neq \emptyset ,$ and thus $B_{\rho }(x+a_{\unicode{x3bb} }^{\unicode{x3bb} '})\subset V_{\rho +\Delta \rho +\Delta _1\rho }(E(\unicode{x3bb} '))=A(\unicode{x3bb} ').$

This shows that $V_{\Delta \rho }(\hat {E}(\unicode{x3bb} ))\subset A^*(\unicode{x3bb} )$ . Applying Proposition 4.1 to $A(\unicode{x3bb} )$ gives

$$ \begin{align*}\sum_{\unicode{x3bb} \in \Lambda}p^{\unicode{x3bb}}\nu(V_{\Delta \rho}(\hat{E}(\unicode{x3bb}))) &\leq \sum_{\unicode{x3bb} \in \Lambda}p^{\unicode{x3bb}}\nu(A^*(\unicode{x3bb})) \leq k_1 \sum_{\unicode{x3bb} \in \Lambda}p^{\unicode{x3bb}}\nu(V_{\rho+\Delta\rho+\Delta_1\rho}(E(\unicode{x3bb}))) \\ &\leq k_1\bigg(1+\frac{1+\Delta_1}{\Delta}\bigg)^{2n-1} \sum_{\unicode{x3bb} \in \Lambda}p^{\unicode{x3bb}}\nu(V_{\Delta\rho}(E(\unicode{x3bb})))\\ &\leq k_4 \sum_{\unicode{x3bb} \in \Lambda}p^{\unicode{x3bb}}\nu(V_{\Delta \rho}(E(\unicode{x3bb}))).\\[-3.8pc] \end{align*} $$

4.2 Proof of Theorem A

In this subsection, we will prove the multidimensional conformal scale recurrence lemma. First we will fix the values of the main parameters playing a role in the proof. This is done to make it clear that there are no contradictions between their values.

Start by choosing a positive constant $\mu $ such that $-\log r^{\underline {\theta }}_{\underline {c}}<\mu $ for any $\underline {\theta }\in \Sigma _j^{-}$ , and $\underline {c}=(c_0,c_1)\in \Sigma _j^\textit{fin}$ a finite sequence with only two symbols. The choice of $\mu $ and the equation $r^{\underline {\theta }}_{\underline {b}\underline {c}}=r^{\underline {\theta }}_{\underline {b}}\cdot r^{\underline {\theta }\underline {b}}_{\underline {c}}$ imply that $\log r^{\underline {\theta }}_{\underline {b}\underline {c}}>\log r^{\underline {\theta }}_{\underline {b}}-\mu .$ This is saying that as the length of $\underline {b}$ increases, the number $\log r^{\underline {\theta }}_{\underline {b}}$ decreases by steps no bigger than $\mu $ .

Now choose $c>0$ such that $c^{-1}\mathrm {diam}(G(\underline {a}^j))\leq \mathrm {diam}(G^{\underline {\theta }^j}(\underline {a}^j)) \leq c\, \mathrm {diam}(G(\underline {a}^j))$ for $\underline {a}^j\in \Sigma _j^\textit{fin}$ , $\underline {\theta }^j \in \Sigma _j^-$ . These constants only depend on $K_1,\ldots ,K_n$ . Fix $\tilde {c}_0>0$ such that

(7) $$ \begin{align} 2\log(c \tilde{c}_0)>\mu. \end{align} $$

We will use Proposition 4.1 with the following data:

$$ \begin{align*} \Lambda&=\Sigma_1 (\tilde{c}_0,\rho)\times\cdots\times \Sigma_n (\tilde{c}_0,\rho),\\ A&=\mathbb{A}_1\times \cdots\times \mathbb{A}_n,\\ \alpha(\underline{a}^1,\ldots,\underline{a}^n)&=(a^1_0,\ldots,a^n_0),\\ \omega(\underline{a}^1,\ldots,\underline{a}^n)&=(a^1_{m_1},\ldots,a^n_{m_n}),\\ a_{\unicode{x3bb}}^{\unicode{x3bb}'}&=\bigg(\log \frac{r_{\underline{b}^1}^{\underline{\theta}^1}}{r_{\underline{b}^n}^{\underline{\theta}^n}},\ldots,\log \frac{r_{\underline{b}^{n-1}}^{\underline{\theta}^{n-1}}}{r_{\underline{b}^n}^{\underline{\theta}^n}},v_{\underline{b}^1}^{\underline{\theta}^1},\ldots,v_{\underline{b}^n}^{\underline{\theta}^n}\bigg), \end{align*} $$

where $\underline {a}^j=(a^j_0,\ldots ,a^j_{m_j})$ , $\unicode{x3bb} =(\underline {a}^1,\ldots ,\underline {a}^n)$ , $\unicode{x3bb} '=(\underline {b}^1,\ldots ,\underline {b}^n)$ , and $\underline {\theta }^j \in \Sigma _j^-$ finishes in $\underline {a}^j$ , $1\leq j\leq n$ . (For every $\underline {a}^j\in \Sigma _j^\textit{fin}$ , we choose, arbitrarily, an element $\underline {\theta }^j\in \Sigma _j^-$ that ends in $\underline {a}^j$ . Using this, we define $a_{\unicode{x3bb} }^{\unicode{x3bb} '}$ .) Assume that the hypothesis of Proposition 4.1 holds, namely that there exist $\Delta _0$ , $k_0$ such that $\| T_{\xi }\|\leq k_0$ for all $|\xi |\in [1,\Delta _0\rho ^{-1}]$ ; this will be verified in the next subsection. Applying the proposition in this setting gives constants $k_1$ , $\tau $ , $\epsilon $ .

Now fix $k_4,k_5>0$ such that $k_1<k_4<k_5<1$ and $\delta>0$ such that

(8) $$ \begin{align} k_4+2\cdot 3^n k_1^{-1} LC_2^{-1}C_4\delta<k_5, \end{align} $$

where $C_2>0$ , $L>0$ are constants such that

(9) $$ \begin{align} L^{-1}\rho^{-(d_1+\cdots+d_n)}&\leq \# \Lambda_i \leq L \rho^{-(d_1+\cdots+d_n)}, \end{align} $$

(10) $$ \begin{align} C_2\rho^{d_1+\cdots+d_n}&\leq p^{\unicode{x3bb}}, \end{align} $$

for any $i\in A$ , $\unicode{x3bb} \in \Lambda $ , and $C_4>0$ is defined by equation (19). All these constants depend only on $\tilde {c}_0$ .

Fix $r>0$ such that

(11) $$ \begin{align} 2r>\delta^{-1}\bigg(9+\frac{1}{4}\bigg)\log (c\tilde{c}_0). \end{align} $$

The choice of $\mu $ and $\tilde {c}_0$ allow us to find $\rho _{1}>0$ , small enough, such that for any family of intervals $I_1,\ldots , I_{n-1}$ , with $\mathrm {diam}(I_j)\geq 2 \log (c \tilde {c}_0)$ , any $x=(t,v)\in J$ with $\mathrm {dist}(t_j,I_j)\leq \delta ^{-1}(9+\tfrac 14)\log (c\tilde {c}_0)$ , and any $\unicode{x3bb} \in \Lambda $ , there exists $\unicode{x3bb} _0=(\underline {b}^1,\ldots ,\underline {b}^n) \in \Sigma _1^\textit{fin}\hspace{-0.5pt}\times\hspace{-0.5pt} \cdots \hspace{-0.5pt}\times\hspace{-0.5pt} \Sigma _n^\textit{fin}$ with the property $x+a_{\unicode{x3bb} }^{\unicode{x3bb} _0}\in I_1\times \cdots \times I_{n-1}\times \mathbb {T}^{n},$ and $\mathrm {diam} (G({\underline {b}^j}))>\rho _1$ , $1\leq j\leq n$ . Choose $c_0>\tilde {c}_0$ big enough such that

for all $\unicode{x3bb} _0=(\underline {b}^1,\ldots ,\underline {b}^n)$ , such that $\mathrm {diam} (G({\underline {b}^j}))>\rho _1$ , $1\leq j\leq n$ , and any $\unicode{x3bb} \in \Lambda $ with $\omega (\unicode{x3bb} _0)=\alpha (\unicode{x3bb} )$ . (If $\unicode{x3bb} =(\underline {a}^1,\ldots ,\underline {a}^n)$ , $\unicode{x3bb} _0=(\underline {b}^1,\ldots ,\underline {b}^n)$ , with $\omega (\unicode{x3bb} _0)=\alpha (\unicode{x3bb} )$ , then $\unicode{x3bb} _0\unicode{x3bb} =(\underline {b}^1\underline {a}^1,\ldots ,\underline {b}^n\underline {a}^n)$ .)

We also fix a constant $\Delta _1>0$ that should be big enough to verify equation (15); this is a condition that only depends on $c_0$ . Finally, we choose $\Delta $ , $\epsilon _1$ as in Corollary 4.1.

Notice that $\Lambda '=\Sigma _1(c_0,\rho )\times \cdots \times \Sigma _n(c_0,\rho )$ contains $\Lambda $ . Choose a function $\varphi :\Lambda '\to \Lambda $ such that we have the following properties.

1. (a) If $\unicode{x3bb} =(\underline {a}^1,\ldots ,\underline {a}^n)\in \Lambda '$ and $\varphi (\unicode{x3bb} )=(\underline {b}^1,\ldots ,\underline {b}^n)$ , then either $\underline {a}^j$ ends with $\underline {b}^j$ or $\underline {b}^j$ ends with $\underline {a}^j$ for every $1\leq j\leq n$ .

2. (b) $\varphi (\unicode{x3bb} )=\unicode{x3bb} \ \text { for all } \unicode{x3bb} \in \Lambda $ .

Thanks to properties (a),(b) of $\varphi $ , there are constants $T_1$ , $T_2$ depending only on $c_0,\, \tilde {c}_0$ , and not on $\rho $ , such that $1\leq T_1\leq \#\varphi ^{-1}(\unicode{x3bb} )\leq T_2 \text { for all } \unicode{x3bb} \in \Lambda .$

We show that we can suppose $F(\unicode{x3bb} )=F(\varphi (\unicode{x3bb} ))$ . Assume that for the given values of $c_0$ and r, there exist $c_1,c_2,c_3,\rho _0>0$ such that the scale recurrence lemma is verified in the special case when $F(\unicode{x3bb} )=F(\varphi (\unicode{x3bb} )) \text { for all } \unicode{x3bb} \in \Lambda '.$ We find new values for $c_1,c_2,c_3, \rho _0>0$ verifying the lemma in the general case. In fact, we do not need to change $c_2,c_3,\rho _0>0$ , just redefine $c_1$ as $c_1/T_2$ . Given a family $\{F(\unicode{x3bb} )\}_{\unicode{x3bb} \in \Lambda '}$ with $\nu (J_r\setminus F(\unicode{x3bb} ))\leq c_1/T_2$ , consider $\tilde {F}(\unicode{x3bb} )=\bigcap _{\unicode{x3bb} '\in \,\varphi ^{-1}(\varphi (\unicode{x3bb} ))}F(\unicode{x3bb} ').$ This new family verifies $\tilde {F}(\unicode{x3bb} )=\tilde {F}(\varphi (\unicode{x3bb} ))$ , moreover, $\nu (J_r\setminus \tilde {F}(\unicode{x3bb} ))\leq c_1$ , then there exists $\tilde {F}^*(\unicode{x3bb} )$ with the properties of the scale recurrence lemma. Taking $F^*(\unicode{x3bb} )=\tilde {F}^*(\unicode{x3bb} )$ gives the lemma in the general case.

For the scale recurrence lemma to hold, it is enough to prove the following statement.

The difference between Statement 4.1 and the scale recurrence lemma is that $\Lambda $ is parameterizing the sets $F(\unicode{x3bb} )$ instead of $\Lambda '$ ; however, $\Lambda '$ , which is much bigger than $\Lambda $ , still parameterizes the set of renormalization operators.

Let $\{F(\unicode{x3bb} )\}_{\unicode{x3bb} \in \Lambda '}$ be a family of sets as in the scale recurrence lemma. We can suppose that $F(\unicode{x3bb} )=F(\varphi (\unicode{x3bb} ))$ . Now assume that Statement 4.1 holds, then we can apply it to the restricted family $\{F(\unicode{x3bb} )\}_{\unicode{x3bb} \in \Lambda }$ , which produces another family $\{F^*(\unicode{x3bb} )\}_{\unicode{x3bb} \in \Lambda }$ . We extend it to $\unicode{x3bb} \in \Lambda '$ by $F^*(\unicode{x3bb} )=F^*(\varphi (\unicode{x3bb} ))$ . It is easily seen that $\{F^*(\unicode{x3bb} )\}_{\unicode{x3bb} \in \Lambda '}$ verifies the desired properties on the scale recurrence lemma. From now on, we will focus in the proof of Statement 4.1.

Suppose we have a family of sets $\{F(\unicode{x3bb} )\}_{\unicode{x3bb} \in \Lambda }$ . Define $E_0(\unicode{x3bb} )=J_r\setminus V_{\Delta \rho }(F(\unicode{x3bb} ))$ , $\unicode{x3bb} \in \Lambda .$ Now we define recursively two families of sets $\{E_m(\unicode{x3bb} )\}_{\unicode{x3bb} \in \Lambda }$ and $\{\tilde {E}_m(\unicode{x3bb} )\}_{\unicode{x3bb} \in \Lambda }$ . The set $\tilde {E}_m(\unicode{x3bb} )$ is given by the $x\in J_r$ such that (when $\unicode{x3bb} '\in \Lambda '\setminus \Lambda $ , the element $a_{\unicode{x3bb} }^{\unicode{x3bb} '}$ is defined in the same way as when $\unicode{x3bb} ,\unicode{x3bb} ' \in \Lambda $ )

$$ \begin{align*}\#\{\unicode{x3bb}'\in \Lambda':\alpha(\unicode{x3bb}')=\omega(\unicode{x3bb}); B_{\Delta_1\rho}(x+a_{\unicode{x3bb}}^{\unicode{x3bb}'})\subset J_r\setminus E_m(\varphi(\unicode{x3bb}'))\}\leq c_3\rho^{-(d_1+\cdots+d_n)},\end{align*} $$

and $E_{m+1}(\unicode{x3bb} )=E_0(\unicode{x3bb} )\cup \tilde {E}_m(\unicode{x3bb} )$ . (In fact, since $\tilde {E}_j(\unicode{x3bb} ) \subset \tilde {E}_{j+1}(\unicode{x3bb} )$ , we have $E_m(\unicode{x3bb} )=E_0(\unicode{x3bb} )\cup \tilde {E}_0(\unicode{x3bb} )\cup \cdots \cup \tilde {E}_{m-1}(\unicode{x3bb} )$ .) The value of $c_3$ will be fixed during the proof of the next lemma. Note that for $x\in \tilde {E}_0(\unicode{x3bb} )$ , one has

$$ \begin{align*}\{\unicode{x3bb}'\in \Lambda':\,B_{\Delta_1\rho}(x+a_{\unicode{x3bb}}^{\unicode{x3bb}'})&\subset J_r\setminus E_1(\varphi(\unicode{x3bb}'))\}\\ &\subset \{\unicode{x3bb}'\in \Lambda':\,B_{\Delta_1\rho}(x+a_{\unicode{x3bb}}^{\unicode{x3bb}'})\subset J_r \setminus E_0(\varphi(\unicode{x3bb}'))\},\end{align*} $$

and hence $\tilde {E}_0(\unicode{x3bb} )\subset \tilde {E}_1(\unicode{x3bb} )$ . Analogously, one proves that $\tilde {E}_m(\unicode{x3bb} )\subset \tilde {E}_{m+1}(\unicode{x3bb} )$ , $E_m(\unicode{x3bb} )\subset E_{m+1}(\unicode{x3bb} )$ for all m.

Lemma 4.1. If $c_1$ , $c_3$ , $\rho _0$ are sufficiently small, then

(12) $$ \begin{align} \sum_{\unicode{x3bb} \in \Lambda}p^{\unicode{x3bb}}\nu(V_{\Delta \rho}(\tilde{E}_m(\unicode{x3bb})))&\leq k_5 \sum_{\unicode{x3bb} \in \Lambda}p^{\unicode{x3bb}}\nu(V_{\Delta \rho}(E_m(\unicode{x3bb}))), \end{align} $$

(13) $$ \begin{align} \sum_{\unicode{x3bb} \in \Lambda}p^{\unicode{x3bb}}\nu(V_{\Delta \rho}(E_m(\unicode{x3bb}))) &\leq \frac{\nu(J_{r+\Delta \rho}\setminus J_r)+c_1}{1-k_5}. \end{align} $$

Before proving the lemma, we will use it to prove Statement 4.1. Consider $E_{\infty }(\unicode{x3bb} )=\bigcup _{m\geq 0}E_m(\unicode{x3bb} )$ ; thanks to equation (13), we have

(14) $$ \begin{align}\sum_{\unicode{x3bb} \in \Lambda}p^{\unicode{x3bb}}\nu(V_{\Delta \rho}(E_{\infty}(\unicode{x3bb}))) \leq \frac{\nu(J_{r+\Delta \rho}\setminus J_r)+c_1}{1-k_5}.\end{align} $$

Now define $F^*(\unicode{x3bb} )=J_r\setminus E_{\infty }(\unicode{x3bb} )$ . We will prove that this family of sets has the desired properties.

(i) Since $E_0(\unicode{x3bb} )\subset E_{\infty }(\unicode{x3bb} )$ then $F^*(\unicode{x3bb} )\subset J_r\setminus E_0(\unicode{x3bb} )=V_{\Delta \rho }(F(\unicode{x3bb} ))$ , choosing $c_2=\Delta $ gives the first property.

(ii) Let $x\in F^*(\unicode{x3bb} )$ , then $x\notin \tilde {E}_m(\unicode{x3bb} )$ and the set

$$ \begin{align*}A_m=\{\unicode{x3bb}'\in \Lambda': \alpha(\unicode{x3bb}')=\omega(\unicode{x3bb});\,\,B_{\Delta_1 \rho}(x+a_{\unicode{x3bb}}^{\unicode{x3bb}'})\subset J_r\setminus E_m(\varphi(\unicode{x3bb}'))\}\end{align*} $$

has more than $c_3\rho ^{-(d_1+\cdots +d_n)}$ elements for all m. Moreover, since $E_m(\varphi (\unicode{x3bb} '))\subset E_{m+1}(\varphi (\unicode{x3bb} '))$ , one has $A_{m+1}\subset A_m$ and then $\#(\bigcap _{m\geq 0}A_m)\geq c_3\rho ^{-(d_1+\cdots +d_n)}.$ Therefore,

$$ \begin{align*}\#\{\unicode{x3bb}'\in \Lambda': \alpha(\unicode{x3bb}')=\omega(\unicode{x3bb});\,\,B_{\Delta_1 \rho}(x+a_{\unicode{x3bb}}^{\unicode{x3bb}'})\subset F^*(\varphi(\unicode{x3bb}'))\}\geq c_3\rho^{-(d_1+\cdots+d_n)}.\end{align*} $$

To finish, it is enough to prove that for any $\tilde {\underline {\theta }}^1,\ldots,\tilde {\underline {\theta }}^n $ such that $\tilde {\underline {\theta }}^j\wedge \underline {a}^j\in \Sigma _j(c_0,\rho )$ , $1\leq j\leq n$ , one has

$$ \begin{align*}B_{\rho}\bigg(x+\bigg(\log \frac{r_{\underline{b}^1}^{\tilde{\underline{\theta}}^1}}{r_{\underline{b}^n}^{\tilde{\underline{\theta}}^n}},\ldots,\log \frac{r_{\underline{b}^{n-1}}^{\tilde{\underline{\theta}}^{n-1}}}{r_{\underline{b}^n}^{\tilde{\underline{\theta}}^n}},v_{\underline{b}^1}^{\tilde{\underline{\theta}}^1},\ldots,v_{\underline{b}^n}^{\tilde{\underline{\theta}}^n}\bigg)\bigg)\subset B_{\Delta_1 \rho}(x+a_{\unicode{x3bb}}^{\unicode{x3bb}'}),\end{align*} $$

where $\unicode{x3bb} '=(\underline {b}^1,\ldots ,\underline {b}^n)$ , $\unicode{x3bb} =(\underline {a}^1,\ldots ,\underline {a}^n)$ , $a_{\unicode{x3bb} }^{\unicode{x3bb} '}=(\log ({r_{\underline {b}^1}^{\underline {\theta }^1}}/{r_{\underline {b}^n}^{\underline {\theta }^n}}),\ldots ,\log ({r_{\underline {b}^{n-1}}^{\underline {\theta }^{n-1}}}/ {r_{\underline {b}^n}^{\underline {\theta }^n}}), v_{\underline {b}^1}^{\underline {\theta }^1},\ldots ,v_{\underline {b}^n}^{\underline {\theta }^n})$ for some $\underline {\theta }^j$ ending in $\underline {a}^j$ , $1\leq j\leq n$ . This is accomplished by taking $\Delta _1$ big. More precisely, since

$$ \begin{align*} \|D(k^{\underline{\theta}^j}\circ (k^{\tilde{\underline{\theta}}^j})^{-1})(z)-I\|\leq C \mathrm{diam} (G(\underline{\theta}^j\wedge \tilde{\underline{\theta}}^j))\leq \tilde{C}\rho\end{align*} $$

for all $1\leq j\leq n$ , for some constant $\tilde {C}$ only depending on $c_0$ , we can conclude that $|\! \log r_{\underline {b}}^{\underline {\theta }^j}-\log r_{\underline {b}}^{\tilde {\underline {\theta }}^j}|\leq \tilde {C}_1 \rho ,\,\,\,| v_{\underline {b}}^{\underline {\theta }^j}-v_{\underline {b}}^{\tilde {\underline {\theta }}^j}|\leq \tilde {C}_1 \rho $ for all $1\leq j\leq n$ , for some constant $\tilde {C}_1$ only depending on $c_0$ . Therefore, imposing

(15) $$ \begin{align} 1+2(2n-1)\tilde{C}_1<\Delta_1 \end{align} $$

would be sufficient to guarantee the second property.

(iii) By equation (14), choosing $c_1$ , $\rho _0$ small such that

$$ \begin{align*}\frac{\nu(J_{r+\Delta \rho}\setminus J_r)+c_1}{1-k_5}<\frac{C_1T_1}{2(T_1+T_2)}\nu(J_r),\end{align*} $$

where $C_1$ is a constant such that $p^{\unicode{x3bb} }\geq C_1 (\#\Lambda )^{-1}\ \text { for all } \unicode{x3bb} \in \Lambda $ , implies that

$$ \begin{align*}\sum_{\unicode{x3bb} \in\Lambda} p^{\unicode{x3bb}}\nu(F^*(\unicode{x3bb}))\geq \bigg(1-\frac{C_1T_1}{2(T_1+T_2)}\bigg)\nu(J_r).\end{align*} $$

Let $A=\{\unicode{x3bb} : \nu (F^*(\unicode{x3bb} ))\geq \nu (J_r)/2\}$ , and hence,

$$ \begin{align*} \bigg(1-\frac{C_1T_1}{2(T_1+T_2)}\bigg)\nu(J_r) &\leq \sum_{\unicode{x3bb} \in A} p^{\unicode{x3bb}}\nu(F^*(\unicode{x3bb})) + \sum_{\unicode{x3bb} \in \Lambda\setminus A} p^{\unicode{x3bb}}\nu(F^*(\unicode{x3bb}))\\ &\leq \nu(J_r) \sum_{\unicode{x3bb} \in A} p^{\unicode{x3bb}} + \frac{\nu(J_r)}{2} \sum_{\unicode{x3bb} \in \Lambda\setminus A} p^{\unicode{x3bb}}\\ &= \nu(J_r)-\frac{\nu(J_r)}{2} \sum_{\unicode{x3bb} \in \Lambda\setminus A} p^{\unicode{x3bb}}. \end{align*} $$

From this inequality, we get $({C_1}/{2})({\#(\Lambda \setminus A)}/{\#\Lambda })\leq \tfrac 12 \sum _{\unicode{x3bb} \in \Lambda \setminus A} p^{\unicode{x3bb} } \leq {C_1T_1}/ {2(T_1+T_2)}.$ Finally, this implies $\# A>(T_2/(T_1+T_2))\#\Lambda $ , as we wanted.

Proof of Lemma 4.1

Choose $c_3>0$ , $\epsilon _2>0$ small such that

(16) $$ \begin{align} \bigg( c_3+ C_2^{-1}\frac{\epsilon_2}{\epsilon_1}\bigg)\rho^{-(d_1+\cdots+d_n)}<(1-\tau)N_{\omega(\unicode{x3bb})}\quad\text{for all } \unicode{x3bb} \in \Lambda, \end{align} $$

where $C_2$ is a constant such that $p^{\unicode{x3bb} }\geq C_2 \rho ^{d_1+\cdots +d_n}\ \text { for all } \unicode{x3bb} \in \Lambda $ . We suppose that $c_1$ , $\rho _0$ are small enough such that

(17) $$ \begin{align} \frac{\nu(J_{r+\Delta \rho}\setminus J_r)+c_1}{1-k_5}<\epsilon_2. \end{align} $$

We will proceed by induction following the scheme

$$ \begin{align*} \text{equation }({13})\text{ for}\ m \Rightarrow \text{equation }({12})\text{ for}\ m \Rightarrow \text{equation }({13})\text{ for}\ m+1. \end{align*} $$

For the base case, equation (13) for $m=0$ , notice that $V_{\Delta \rho }(E_0(\unicode{x3bb} ))\subset J_{r+\Delta \rho }\setminus F(\unicode{x3bb} ).$ Therefore,

$$ \begin{align*}\nu(V_{\Delta \rho}(E_0(\unicode{x3bb}))) \leq \nu(J_{r+\Delta \rho}\setminus J_r)+\nu(J_r \setminus F(\unicode{x3bb}))\leq \frac{\nu(J_{r+\Delta \rho}\setminus J_r)+c_1}{1-k_5}.\end{align*} $$

Now we prove ‘equation (13) for $m\ \Rightarrow $ equation (12) for m’: define $\Lambda _b=\{\unicode{x3bb} \in \Lambda : \nu (V_{\Delta \rho }(E_m(\unicode{x3bb} )))<\epsilon _1\}$ , $A(\unicode{x3bb} )=E_m(\unicode{x3bb} )$ for $\unicode{x3bb} \in \Lambda _b$ and $A(\unicode{x3bb} )=\emptyset $ otherwise. We will show that $\tilde {E}_m(\unicode{x3bb} )\cap J_{r-2\log (c\tilde {c}_0)-\Delta _1\rho }\subset \hat {A}(\unicode{x3bb} ).$ Here $\hat {A}(\unicode{x3bb} )$ is the set from Corollary 4.1, that is, the $x\in J$ such that

$$ \begin{align*}\frac{1}{\# \Lambda_{\omega(\unicode{x3bb})}}\cdot \#\{\unicode{x3bb}'\in \Lambda_{\omega(\unicode{x3bb})}:B_{\Delta_1 \rho}(x+a_{\unicode{x3bb}}^{\unicode{x3bb}'})\subset J\setminus A(\unicode{x3bb}') \} \leq 1-\tau.\end{align*} $$

Using equation (17) gives $\epsilon _2>\sum _{\unicode{x3bb} \in \Lambda }p^{\unicode{x3bb} }\nu (V_{\Delta \rho }(E_m(\unicode{x3bb} )))\geq C_2 \rho ^{d_1+\cdots +d_n}\epsilon _1 \#(\Lambda \setminus \Lambda _b),$ and hence,

$$ \begin{align*}\#(\Lambda\setminus\Lambda_b)\leq C_2^{-1}\frac{\epsilon_2}{\epsilon_1}\rho^{-(d_1+\cdots+d_n)}.\end{align*} $$

Let $x\in \tilde {E}_m(\unicode{x3bb} )\cap J_{r-2\log (c\tilde {c}_0)-\Delta _1\rho }$ , using the last inequality, equation (16), and the fact that $a_{\unicode{x3bb} }^{\unicode{x3bb} '}\in J_{2\log (c \tilde {c}_0)}\ \text { for all } \unicode{x3bb} , \unicode{x3bb} '\in \Lambda $ , we have

$$ \begin{align*} \#\{\unicode{x3bb}'\in \Lambda_{\omega(\unicode{x3bb})}:B_{\Delta_1 \rho}(x+a_{\unicode{x3bb}}^{\unicode{x3bb}'})&\subset J\setminus A(\unicode{x3bb}') \} \\ &\leq \#\{\unicode{x3bb}'\in \Lambda_b:B_{\Delta_1 \rho}(x+a_{\unicode{x3bb}}^{\unicode{x3bb}'})\subset J\setminus E_m(\varphi(\unicode{x3bb}')) \}\\ &\quad+ \#(\Lambda\setminus \Lambda_b)\\ &\leq \#\{\unicode{x3bb}'\in \Lambda':B_{\Delta_1 \rho}(x+a_{\unicode{x3bb}}^{\unicode{x3bb}'})\subset J_r\setminus E_m(\varphi(\unicode{x3bb}'))\}\\ &\quad+C_2^{-1}\frac{\epsilon_2}{\epsilon_1}\rho^{-(d_1+\cdots+d_n)}\\ &\leq (c_3+C_2^{-1}\frac{\epsilon_2}{\epsilon_1})\rho^{-(d_1+\cdots+d_n)}\\ &<(1-\tau)N_{\omega(\unicode{x3bb})}. \end{align*} $$

Hence, $x\in \hat {A}(\unicode{x3bb} )$ , and we have shown $\tilde {E}_m(\unicode{x3bb} )\cap J_{r-2\log (c\tilde {c}_0)-\Delta _1\rho }\subset \hat {A}(\unicode{x3bb} ).$ Clearly the family $A(\unicode{x3bb} )$ satisfies the hypothesis of Corollary 4.1, that is, $\nu (V_{\Delta \rho }(A(\unicode{x3bb} )))<\epsilon _1$ . Using the corollary gives

(18) $$ \begin{align} \sum_{\unicode{x3bb}\in\Lambda}p^{\unicode{x3bb}}\nu(V_{\Delta\rho}(\tilde{E}_m(\unicode{x3bb})\cap J_{r-2\log (c\tilde{c}_0)-\Delta_1\rho}))&\leq \sum_{\unicode{x3bb}\in\Lambda}p^{\unicode{x3bb}}\nu(V_{\Delta\rho}(\hat{A}(\unicode{x3bb})))\nonumber \\ &\leq k_4 \sum_{\unicode{x3bb}\in\Lambda}p^{\unicode{x3bb}}\nu(V_{\Delta\rho}(A(\unicode{x3bb})))\nonumber \\ &\leq k_4\sum_{\unicode{x3bb}\in\Lambda}p^{\unicode{x3bb}}\nu(V_{\Delta\rho}(E_m(\unicode{x3bb}))). \end{align} $$

Now we estimate $\tilde {E}_m(\unicode{x3bb} )$ outside of $J_{r-2\log (c\tilde {c}_0)-\Delta _1\rho }$ . Assume that $\rho _0$ is small enough such that $2\Delta \rho <\tfrac 14\log (c\tilde {c}_0)$ , $\Delta _1 \rho <\tfrac 14\log (c\tilde {c}_0)$ . By equation (11), for every $j=1,\ldots ,n$ , there exist intervals

$$ \begin{align*} I_j^- &\subset [-r,-r+\delta^{-1}(9\log (c\tilde{c}_0)+2\Delta \rho)]\subset [-r,r],\\ I_j^+ &\subset [r-\delta^{-1}(9\log (c\tilde{c}_0)+2\Delta \rho),r]\subset [-r,r], \end{align*} $$

with $\mathrm {diam}(I_j^-)=\mathrm {diam}(I_j^+)=9\log (c\tilde {c}_0)$ such that

$$ \begin{align*} \sum_{\unicode{x3bb}\in \Lambda}p^{\unicode{x3bb}}\nu(V_{\Delta \rho}(E_m(\unicode{x3bb})\cap J^-_j))&\leq \delta \sum_{\unicode{x3bb}\in \Lambda}p^{\unicode{x3bb}}\nu(V_{\Delta \rho}(E_m(\unicode{x3bb}))),\\ \sum_{\unicode{x3bb}\in \Lambda}p^{\unicode{x3bb}}\nu(V_{\Delta \rho}(E_m(\unicode{x3bb})\cap J^+_j))&\leq \delta \sum_{\unicode{x3bb}\in \Lambda}p^{\unicode{x3bb}}\nu(V_{\Delta \rho}(E_m(\unicode{x3bb}))), \end{align*} $$

where $J^{+(-)}_j=\{(t_1,\ldots ,t_{n-1},v)\in J_r:t_j\in I^{+(-)}_j\}$ .

For every pair $(A,B)$ such that $A,B \subset \{1,\ldots ,n-1\}$ and $A\cap B=\emptyset $ , consider $y_{A,B}=(t_1(A,B),\ldots ,t_{n-1}(A,B),0)\in J$ given by

$$ \begin{align*} t_j(A,B)= \begin{cases} -r+\log(c\tilde{c}_0) & \mbox{if } j \in A, \\ r-\log(c\tilde{c}_0) & \mbox{if } j \in B, \\ 0 & \mbox{otherwise}. \end{cases} \end{align*} $$

We also define sets $\tilde {J}_{A,B}, J_{A,B} \subset J$ in the following way:

$$ \begin{align*} J_{A,B}=\{(t_1,\ldots,t_{n-1},v)\in J_{r}:t_j\in I^{+}_j\,& \mbox{if } j\in B, t_j\in I^{-}_j\, \mbox{if } j\in A\},& \\ \tilde{J}_{A,B}=\{(t_1,\ldots,t_{n-1},v)\in J_r:t_j\in \tilde{I}^{+}_j\,& \mbox{if } j\in B, t_j\in \tilde{I}^{-}_j\, \mbox{if } j\in A\}, \end{align*} $$

where the interval $\tilde {I}^{u}_j$ has the same center as $I^{u}_j$ and has length $2\log (c\tilde {c}_0)$ ; here $u=+$ or $-$ .

Fix $\unicode{x3bb} \in \Lambda $ . From the choice of $\rho _1$ and $c_0$ , we know that there is $\unicode{x3bb} _{A,B}\in \Sigma _1^\textit{fin}\times \cdots \times \Sigma _n^\textit{fin}$ such that $y_{A,B}+a_{\unicode{x3bb} }^{\unicode{x3bb} _{A,B}}\in \tilde {J}_{A,B},$ and $\unicode{x3bb} _{A,B}\unicode{x3bb} '\in \Lambda '\ \text { for all } \unicode{x3bb} '\in \Lambda _{\omega (\unicode{x3bb} _{A,B})}$ .

Let $x\in V_{\Delta \rho }(\tilde {E}_m(\unicode{x3bb} )\cap J_r\setminus J_{r-2\log (c\tilde {c}_0)-\Delta _1 \rho }),$ then there is $y\in \tilde {E}_m(\unicode{x3bb} )\cap J_r\setminus J_{r-2\log (c\tilde {c}_0)-\Delta _1 \rho }$ , $y\in B_{\Delta \rho }(x)$ , and for y, we have

$$ \begin{align*}\#\{\unicode{x3bb}'\in\Lambda': B_{\Delta_1\rho}(y+a_{\unicode{x3bb}}^{\unicode{x3bb}'})\subset J_r\setminus E_m(\varphi(\unicode{x3bb}'))\}<c_3\rho^{-(d_1+\cdots+d_n)}.\end{align*} $$

Write $y=(t_1,\ldots ,t_{n-1},v)$ and consider the sets

$$ \begin{align*} A&=\{j\in [1,n-1]\cap \mathbb{Z}: t_j<-r+2\log (c\tilde{c}_0)+\Delta_1 \rho\},\\ B&=\{j\in [1,n-1]\cap \mathbb{Z}: t_j>r-2\log (c\tilde{c}_0)-\Delta_1 \rho\}. \end{align*} $$

Since $y\notin J_{r-2\log (c\tilde {c}_0)-\Delta _1 \rho }$ , we know that $A\cup B\neq \emptyset $ and we can consider $\unicode{x3bb} _{A,B}$ . Given that $\#\Lambda _{\omega (\unicode{x3bb} _{A,B})}\geq L^{-1}\rho ^{-(d_1+\cdots +d_n)}$ , we conclude that

where we are assuming that $c_3<L^{-1}/2$ . Notice that $B_{\Delta _1\rho }(y+a_{\unicode{x3bb} }^{\unicode{x3bb} _{A,B}\unicode{x3bb} '})\cap J_r\subset J_{A,B}\ \text { for all } \unicode{x3bb} '\in \Lambda _{\omega (\unicode{x3bb} _0)}$ ; therefore,

Hence, $V_{\Delta \rho }(\tilde {E}_n(\unicode{x3bb} )\cap J_r\setminus J_{r-2\log (c\tilde {c}_0)-\Delta _1 \rho })$ is contained in

where the first sum is over all pairs $(A,B)$ such that $A,B\subset \{1,\ldots ,n\}$ , $A\cap B=\emptyset $ , $A\cup B\neq \emptyset $ . Now using Chebyshev’s inequality, we get

where $C_4>0$ is a constant, only depending on $\tilde {c}_0$ , such that

(19)

Now we will sum over $\unicode{x3bb} $ . By the definition of $\Delta $ , we know that $(1+{\Delta _1}/{\Delta })^{2n-1}<k_1^{-1},$ and using $p^{\unicode{x3bb} '}\geq C_2\rho ^{d_1+\cdots +d_n}$ , we get

$$ \begin{align*} &\sum_{\unicode{x3bb}\in \Lambda}p^{\unicode{x3bb}}\nu(V_{\Delta\rho}(\tilde{E}_m(\unicode{x3bb})\cap J_r\setminus J_{r-2\log (c\tilde{c}_0)-\Delta_1 \rho}))\\ &\quad\leq 2\bigg(1+\frac{\Delta_1}{\Delta}\bigg)^{2n-1} C_4 L\rho^{d_1+\cdots+d_n}\sum_{(A,B)} \sum_{\unicode{x3bb}'\in \Lambda}\nu(V_{\Delta\rho}(E_m(\unicode{x3bb}')\cap J_{A,B}))\\ &\quad\leq 2k_1^{-1}C_4L\rho^{d_1+\cdots+d_n}\frac{1}{C_2\rho^{d_1+\cdots+d^{\prime}_n}}\sum_{(A,B)} \sum_{\unicode{x3bb}'\in \Lambda}p^{\unicode{x3bb}'}\nu(V_{\Delta\rho}(E_m(\unicode{x3bb}')\cap J_{A,B}))\\ &\quad\leq 2\cdot 3^n k_1^{-1} LC_2^{-1}C_4\delta \sum_{\unicode{x3bb}\in \Lambda}p^{\unicode{x3bb}}\nu(V_{\Delta\rho}(E_m(\unicode{x3bb}))). \end{align*} $$

Putting this inequality together with (18) gives

$$ \begin{align*} \sum_{\unicode{x3bb}\in \Lambda}p^{\unicode{x3bb}}\nu(V_{\Delta\rho}(\tilde{E}_m(\unicode{x3bb})))&\leq (k_4+2\cdot 3^n k_1^{-1} LC_2^{-1}C_4\delta)\sum_{\unicode{x3bb}\in \Lambda}p^{\unicode{x3bb}}\nu(V_{\Delta\rho}(E_m(\unicode{x3bb})))\\ &\leq k_5 \sum_{\unicode{x3bb}\in \Lambda}p^{\unicode{x3bb}}\nu(V_{\Delta\rho}(E_m(\unicode{x3bb}))), \end{align*} $$

which is equation (12) for m. Here we have used equation (8) where $\delta $ was chosen.

To finish, we prove ‘equation (13) for m and equation (12) for $m\ \Rightarrow $ equation (13) for $m+1$ ’. Since $E_{m+1}(\unicode{x3bb} )=E_0(\unicode{x3bb} )\cup \tilde {E}_m(\unicode{x3bb} )$ and $\nu (V_{\Delta \rho }(E_0(\unicode{x3bb} )))\leq \nu (J_{r+\Delta \rho }\setminus J_r)+c_1$ , we get

$$ \begin{align*} \sum_{\unicode{x3bb}\in\Lambda}p^{\unicode{x3bb}}\nu(V_{\Delta\rho}(E_{m+1}(\unicode{x3bb})))&\leq \sum_{\unicode{x3bb}\in\Lambda}p^{\unicode{x3bb}}\nu(V_{\Delta\rho}(E_0(\unicode{x3bb})))+\sum_{\unicode{x3bb}\in\Lambda}p^{\unicode{x3bb}}\nu(V_{\Delta\rho}(\tilde{E}_{m}(\unicode{x3bb})))\\ &\leq \nu(J_{r+\Delta \rho}\setminus J_r)+c_1+k_5\sum_{\unicode{x3bb}\in\Lambda}p^{\unicode{x3bb}}\nu(V_{\Delta\rho}(E_{m}(\unicode{x3bb})))\\ &\leq \nu(J_{r+\Delta \rho}\setminus J_r)+c_1+k_5\frac{\nu(J_{r+\Delta \rho}\setminus J_r)+c_1}{1-k_5}\\ &=\frac{\nu(J_{r+\Delta \rho}\setminus J_r)+c_1}{1-k_5}.\\[-3.7pc] \end{align*} $$

4.3 Proof of hypothesis on Proposition 4.1

In this subsection, we prove that there exist $0<k_0<1$ , $\Delta _0>0$ such that $\|T_{\xi }\|\leq k_0$ for all $|\xi |\in [1,\Delta _0\rho ^{-1}]$ , and these constants does not depend on $\rho $ . Remember that the operator $T_{\xi }:\mathbb {C}^{\Lambda }\to \mathbb {C}^{\Lambda }$ is given by $T_{\xi }((z_{\unicode{x3bb} })_{\unicode{x3bb} \in \Lambda })=(w_{\unicode{x3bb} })_{\unicode{x3bb} \in \Lambda }$ , where $w_{\unicode{x3bb} }=\sum _{\unicode{x3bb} '\in \Lambda } p_{\unicode{x3bb} }^{\unicode{x3bb} '}\xi (a_{\unicode{x3bb} }^{\unicode{x3bb} '})z_{\unicode{x3bb} '}.$ Notice that $\mathbb {C}^{\Lambda }$ can be decomposed in two ways:

$$ \begin{align*}\mathbb{C}^{\Lambda}=\bigoplus_{i\in A} \mathbb{C}^{\Lambda_i},\,\,\,\mathbb{C}^{\Lambda}=\bigoplus_{j\in A} \mathbb{C}^{\Lambda^j},\end{align*} $$

and the operator $T_{\xi }$ sends $\mathbb {C}^{\Lambda _i}$ into $\mathbb {C}^{\Lambda ^i}$ . Let $\|\cdot \|_i$ , $\|\cdot \|^{j}$ be the restriction to $\mathbb {C}^{\Lambda _i}$ , $\mathbb {C}^{\Lambda ^j}$ , respectively, of the norm $\|\cdot \|$ on $\Lambda $ . Note that

$$ \begin{align*}\|z\|^2=\sum_{i\in A}\|\pi_i(z)\|_i^2=\sum_{j\in A}(\|\pi^j(z)\|^j)^2 \text{ for all } z\in \mathbb{C}^{\Lambda},\end{align*} $$

where $\pi _i:\mathbb {C}^{\Lambda }\to \mathbb {C}^{\Lambda _i}$ , $\pi ^j:\mathbb {C}^{\Lambda }\to \mathbb {C}^{\Lambda ^j}$ are the projections given by the decompositions. This implies that $\|T_{\xi }\|\leq k_0$ if and only if $\|T_{\xi }|_{\mathbb {C}^{\Lambda _i}}\|\leq k_0\ \text { for all } i\in A$ , where $T_{\xi }|_{\mathbb {C}^{\Lambda _i}}$ is the restriction $T_{\xi }|_{\mathbb {C}^{\Lambda _i}}:(\mathbb {C}^{\Lambda _i},\|\cdot \|_i)\to (\mathbb {C}^{\Lambda ^i},\|\cdot \|^i).$ We start by supposing that there exist $\rho>0$ , $|\xi |\in [1,\Delta _0\rho ^{-1}]$ , $i\in A$ such that

$$ \begin{align*}\|T_{\xi}|_{\mathbb{C}^{\Lambda_i}}\|\geq (1-\eta_0)^{1/2}.\end{align*} $$

From this, we will derive a series of inequalities depending on parameters $\eta _0$ , $\eta _1$ , $\eta _2, \ldots,$ each new parameter $\eta _{j+1}$ will depend on $\eta _{j}$ , not on $\rho $ , and $\lim _{\eta _j\to 0}\eta _{j+1}=0$ . Finally, we will see that with the appropriate value of $\Delta _0$ , the last $\eta _j$ will be bounded away from zero and then also $\eta _0$ , and this will complete the proof.