2 Equidistribution of rational submanifolds

Let $\mathscr {N}=G/\Lambda $ be a Riemannian compact nilmanifold and m its Haar probability measure. An affine rational submanifold $\mathscr {S}\subseteq \mathscr {N}$ is a quotient $gH/(H\cap \Lambda )$ , where H is a connected closed subgroup of G intersecting $\Lambda $ cocompactly and $g\in G$ is a translation parameter. It carries a unique $gHg^{-1}$ -invariant probability measure, denoted by $m_{\mathscr {S}}$ .

The goal of the section is to reduce quantitative equidistribution of $(\mathscr {S}, m_{\mathscr {S}})$ in $(\mathscr {N}, m)$ to a diophantine condition on the Lie algebra of H.

We call $\mathfrak {h} \subseteq \mathfrak {g}$ the respective Lie algebras of $H\subseteq G$ and write $\mathfrak {t}=\mathfrak {g}/[\mathfrak {g}, \mathfrak {g}]$ for the largest abelian quotient of $\mathfrak {g}$ . The projection map from $\mathfrak {g}$ to $\mathfrak {t}$ is denoted by $\mathfrak {g} \rightarrow \mathfrak {t}, w\mapsto \overline {w}$ and sends $\log \Lambda $ to a lattice $Z_{\Lambda }=\overline {\log \Lambda }$ in $\mathfrak {t}$ . For future reference, we fix a basis of the latter, which leads to identifications $\mathfrak {t}\equiv \mathbb {R}^d$ , $Z_{\Lambda }\equiv \mathbb {Z}^d$ .

Proposition 2.1. (Equidistribution of rational submanifolds)

Let $\mathscr {N}=G/\Lambda $ be a Riemannian compact nilmanifold and let $\theta \in (0,1]$ . There exists $L>0$ such that for every $\delta \in (0,1/2)$ , any affine rational submanifold $\mathscr {S}\subseteq \mathscr {N}$ satisfying

$$ \begin{align*} \min \{\lVert q\rVert : q\in \mathbb{Z}^d\setminus \{0\}, \langle q, \overline{\mathfrak{h}}\rangle =0\} \geq \bigg(\frac{1}{\delta}\bigg)^L \end{align*} $$

is $\delta $ -equidistributed with respect to $\mathcal {H}^\theta (\mathscr {N})$ , that is,

$$ \begin{align*} \bigg|\int_{\mathscr{S}} f dm_{\mathscr{S}} - \int_{\mathscr{N}}f \,dm\bigg| \leq \delta \lVert f\rVert_{\mathcal{H}^\theta} \end{align*} $$

for all $f\in \mathcal {H}^\theta (\mathscr {N})$ .

We deduce this proposition from Theorem 2.2, which is itself a direct corollary of a much stronger result of Green and Tao [Reference Green and Tao11, Theorem 1.16].

The above result does not use the full force of [Reference Green and Tao11, Theorem 1.16] in two significant ways. First, the result of Green and Tao is for general polynomial sequences in place of $g\exp (kw)$ . Second, it is possible to control in a very efficient way how large n needs to be in item (1) at the expense of replacing $\langle q,\overline w\rangle \in \mathbb {Z}$ in item (2) by an upper bound on $\operatorname {\mathrm {dist}}(\langle q,\overline w\rangle ,\mathbb {Z})$ . These features are crucial in some other applications of [Reference Green and Tao11, Theorem 1.16], but we do not need them.

Proof of Proposition 2.1

We suppose without loss of generality that $\theta =1$ (see [Reference Gorodnik and Spatzier10, proof of Theorem 2.3]). We choose $L>0$ as in Theorem 2.2 and consider an affine rational submanifold $\mathscr {S}\subseteq \mathscr {N}$ as well as a constant $\delta \in (0,1/2)$ .

Assume that $\mathscr {S}$ is not $\delta $ -equidistributed for $\mathcal {H}^1(\mathscr {N})$ . Fix a vector $w\in \mathfrak {h}$ such that $\lVert \overline {w}\rVert \leq \tfrac 12\delta ^L$ and whose projection in $\mathfrak {h}/[\mathfrak {h}, \mathfrak {h}]$ does not belong to any translate of a $H\cap \Lambda $ -rational proper subspace by a $H\cap \Lambda $ -rational vector. By Theorem 2.2, or in this case even by an earlier theorem of Leon Green [Reference Auslander, Green and Hahn1, Reference Green12], we have the weak- $*$ convergence:

$$ \begin{align*} \frac{1}{n}\sum_{k=1}^{n}\delta_{g\exp(kw)\Lambda} \longrightarrow m_{\mathscr{S}}. \end{align*} $$

In particular, for large enough n, the sequence $(g\exp (kw)\Lambda )_{0\leq k\leq n}$ does not satisfy $\delta $ -equidistribution for $\mathcal {H}^1(\mathscr {N})$ either. Theorem 2.2 yields some integer vector $q\in \mathbb {Z}^d\setminus \{0\}$ such that $\lVert q\rVert < \delta ^{-L}$ and $\langle q,\overline {w}\rangle \in \mathbb {Z}$ . This forces $\langle q,\overline {w}\rangle =0$ as our choice of w guarantees that $|\langle q,\overline {w}\rangle |\leq 1/2$ . As the smallest rational subspace of $\mathfrak {t}$ containing $\overline {w}$ is $\overline {\mathfrak {h}}$ , we get that $\langle q, \overline {\mathfrak {h}}\rangle =0$ , which concludes the proof.

3 Diophantine estimates

Let K be a number field. We denote by $M_K$ the set of its places, by $M_{K,\infty }$ its Archimedean places, and by $M_{K,f}$ its finite places. We use the convention that we include complex places twice (one for each complex conjugate embedding), and we write $|x|_v$ for $x\in K$ and $v\in M_{K,\infty }$ to denote the usual absolute value arising from the embedding of K in $\mathbb {R}$ or $\mathbb {C}$ associated to v. (This convention is not standard, but it is also not unprecedented, see [Reference Masser14, Ch. 14].) The finite places will not play much of a role in this paper, so we do not specify which convention is used for inclusion in $M_{K,f}$ or for the normalization of the corresponding absolute values. We do insist, however, that the product formula $\prod _{v\in M_K}|x|_v=1$ holds for all $x\in K$ .

The projective height of a tuple $(a_1,\ldots ,a_n)\in K^n$ is denoted by

$$ \begin{align*} H(a_1,\ldots,a_n)=\prod_{v\in M_K} \max_j(|a_j|_v^{1/[K:\mathbb{Q}]}). \end{align*} $$

We note that this quantity is a projective notion, that is, it is invariant under multiplication of each coordinate by the same scalar. This fact is an immediate consequence of the product formula.

The ring of integers in K is denoted by ${\mathcal O}(K)$ and for $h\in \mathbb {R}_{\ge 0}$ , we set

$$ \begin{align*} {\mathcal O}(K)_{h}=\{a\in{\mathcal O}(K):|a|_v\le h\text{ for all }v\in M_{K,\infty}\}. \end{align*} $$

We write ${\mathcal O}(K)^\times $ for the group of (multiplicative) units in ${\mathcal O}(K)$ . For $n\in \mathbb {Z}_{\ge 2}$ and $u=(u_1,\ldots ,u_n)\in ({\mathcal O}(K)^\times )^n$ , we write

$$ \begin{align*} \alpha(u)=\min_{I\subset\{1,\ldots, n\}, |I|\ge 2}H(u_i:i\in I)^{1/(|I|-1)}. \end{align*} $$

In what follows, we consider a generalization of the classical unit equation $u_1+u_2=1$ to be solved for $u_1,u_2\in {\mathcal O}(K)^\times $ . This subject has a rich literature, the recent book [Reference Evertse and Győry5] is a good general reference.

Theorem 3.1. Let K be a number field. For all $\varepsilon>0$ and $n\geq 2$ , there is an (ineffective) constant $r>0$ such that the following holds. Let $u=(u_1,\ldots ,u_n)\in ({\mathcal O}(K)^\times )^n$ and let $a_1,\ldots ,a_n\in {\mathcal O}(K)_{h}$ not all $0$ for some $h\in \mathbb {R}_{\ge 0}$ . Suppose

$$ \begin{align*} a_1u_1+\cdots+a_n u_n=0. \end{align*} $$

Then $h\ge r\alpha (u)^{1-\varepsilon }$ .

We comment on how this result is related to the classical unit equation. Taking $n=3$ , $a_1=a_2=1$ , $a_3=-1$ , $u_3=1$ , Theorem 3.1 implies that the classical unit equation has finitely many solutions, a result that goes back to Siegel, Mahler, and Lang.

Theorem 3.1 is not new. It could be deduced (with a slightly weaker exponent) from a result of Evertse [Reference Evertse4], see also [Reference Evertse and Győry5, Theorem 6.1.1]. We give a short proof based on [Reference Evertse4] for the reader’s convenience.

As a simple application of Dirichlet’s box principle, we also show below that the exponent in the lower bound on h is almost optimal in the sense that $1-\varepsilon $ could not be replaced by a number larger than $1$ .

3.1 The subspace theorem

At the heart of the proof of Theorem 3.1 is W. Schmidt’s subspace theorem which we now recall.

Theorem 3.2. [Reference Bombieri and Gubler2, Corollary 7.2.5]

Let K be a number field and let $\{|.|_{v}, v \in M_{K}\}$ be a system of absolute values on K as above. Let $n\geq 2$ and let V be an n-dimensional vector space over K. For each $v\in M_{K,\infty }$ , let $\Lambda _1^{(v)},\ldots ,\Lambda _n^{(v)}$ be a basis of the dual space $V^*$ . Furthermore, let $\Lambda _1^{(0)},\ldots ,\Lambda _n^{(0)}$ be another basis of $V^*$ . Then for all $\varepsilon>0$ , the solutions of

$$ \begin{align*} \prod_{v\in M_{K,\infty}}\prod_{j=1}^n|\Lambda_j^{(v)}(x)|_v \le H(\Lambda_1^{(0)}(x),\ldots,\Lambda_n^{(0)}(x))^{-\varepsilon} \end{align*} $$

for $x\in V$ with $\Lambda ^{(0)}_j(x)\in {\mathcal O}(K)$ for all $j=1,\ldots ,n$ are contained in a finite union of proper subspaces of V.

The functionals $\Lambda _1^{(0)},\ldots ,\Lambda _n^{(0)}$ can be used to identify V with $K^n$ and using this identification, $\Lambda _j^{(v)}$ can be identified with a linear form for each j and v. Using these identifications, Theorem 3.2 is translated into the form in [Reference Bombieri and Gubler2, Corollary 7.2.5]. In our application that follows, there is no natural choice for an identification between V and $K^n$ , so the above formulation will suit our purposes best.

3.2 Proof of Theorem 3.1

The proof is by induction on n. Suppose first $n=2$ . Let $u_1,u_2\in {\mathcal O}(K)^\times $ and $a_1,a_2\in {\mathcal O}(K)_{h}$ for some $h\in \mathbb {R}_{\ge 0}$ such that $a_{1}, a_{2}\neq 0$ and $a_1u_1+a_2u_2=0$ . We observe that

$$ \begin{align*} (u_1,u_2)=\bigg(\frac{u_1}{a_2}a_2,-\frac{u_1}{a_2}a_1\bigg), \end{align*} $$

hence

$$ \begin{align*} H(u_1,u_2)=H(a_1,a_2)\le (h^{1/[K:\mathbb{Q}]})^{[K:\mathbb{Q}]}=h. \end{align*} $$

Since $\alpha (u_1,u_2)=H(u_1,u_2)$ , this proves our claim with $r=1$ .

Now suppose $n\ge 3$ and that the claim holds for all smaller values of n. Let V be the subspace of $K^n$ of those points that satisfy the equation $x_1+\cdots +x_n=0$ . Observe that

$$ \begin{align*} y:=(a_1u_1,\ldots,a_nu_n)\in V. \end{align*} $$

For each place $v\in M_{K,\infty }$ , let $\Lambda _1^{(v)},\ldots ,\Lambda _{n-1}^{(v)}$ be an enumeration of all but one of the functionals $(x_1,\ldots ,x_n)\mapsto x_j$ so that we omit one of the indices j for which $|u_j|_v$ is maximal. We also set $\Lambda _j^{(0)}(x)=x_j$ , say, for $j=1,\ldots ,n-1$ . Observe that, by the product formula,

$$ \begin{align*} \prod_{v\in M_{K,\infty}}\prod_{j=1}^{n-1} |\Lambda_j^{(v)}(u_1,\ldots, u_n)|_{v} =\prod_{v\in M_{K,\infty}}\frac{\prod_{j=1}^{n}|u_j|_v }{\max_j(|u_j|_v)} =H(u_1,\ldots,u_n)^{-[K:\mathbb{Q}]}. \end{align*} $$

Using $a_1,\ldots ,a_n\in {\mathcal O}(K)_{h}$ and the definition of $\Lambda _j^{(v)}$ , we have

$$ \begin{align*} |\Lambda_j^{(v)}(a_1u_1,\ldots,a_nu_n)|_{v}\le h |\Lambda_j^{(v)}(u_1,\ldots,u_n)|_{v} \end{align*} $$

for all j and v. Therefore,

$$ \begin{align*} \prod_{v\in M_{K,\infty}}\prod_{j=1}^{n-1}|\Lambda_j^{(v)}(y)|_v \le h^{(n-1)[K:\mathbb{Q}]} H(u)^{-[K:\mathbb{Q}]}. \end{align*} $$

We assume $h\le \alpha (u)^{1-\varepsilon }$ as we may, for otherwise, the claim is trivial. Then $h\le H(u)^{(1-\varepsilon )/(n-1)}$ and we observe that

$$ \begin{align*} H(\Lambda_1^{(0)}(y),\ldots,\Lambda_{n-1}^{(0)}(y)) &=H(a_1u_1,\ldots,a_{n-1}u_{n-1})\\ &\le h H(u_1,\ldots,u_{n-1}) \le h H(u) \le H(u)^2. \end{align*} $$

Thus,

$$ \begin{align*} \prod_{v\in M_{K,\infty}}\prod_{j=1}^{n-1}|\Lambda_j^{(v)}(y)|_v \le H(u)^{-\varepsilon[K:\mathbb{Q}]}\le H(\Lambda_1^{(0)}(y),\ldots,\Lambda_{n-1}^{(0)}(y))^{-\varepsilon[K:\mathbb{Q}]/2}. \end{align*} $$

Therefore, the subspace theorem applies and it follows that $(a_1u_1,\ldots , a_nu_n)$ is contained in finitely many proper subspaces of V. In what follows, we use the induction hypothesis to show that for any proper subspace U of V, there is a constant $r=r(U)>0$ such that $h\ge r \alpha (u)^{1-\varepsilon }$ holds whenever there are $a_1,\ldots ,a_n$ and $u_1,\ldots , u_n$ that satisfy the assumptions in the proposition with $y\in U$ . We can then take the minimum of $r(U)$ over all subspaces U that arise through the application of the subspace theorem taking into account all possible choices of the functionals $\Lambda _j^{(v)}$ , of which there are finitely many. Then the claim will hold with this minimum in the role of r.

Let now U be a proper subspace of V with $y\in U$ , and let $b_1,\ldots ,b_n\in {\mathcal O}(K)$ be such that

$$ \begin{align*} b_1x_1+\cdots+b_nx_n=0 \end{align*} $$

for all $(x_1,\ldots , x_n)\in U$ and not all $b_j$ are equal. We also suppose that $a_j\neq 0$ for all j, as we may, for otherwise, we may omit the $0$ coordinates, and the induction hypothesis applies directly. We now observe that

$$ \begin{align*} (b_1-b_n)a_1u_1+\cdots+(b_{n-1}-b_n)a_{n-1}u_{n-1}=0 \end{align*} $$

and that not all of the coefficients $(b_j-b_n)a_j$ vanish.

We also note that $(b_j-b_n)a_j\in {\mathcal O}(K)_{Ch}$ for some constant C depending on the $b_j$ . Therefore, we are in a position to apply the induction hypothesis and conclude

$$ \begin{align*} Ch\ge c\alpha(u_1,\ldots,u_{n-1})^{1-\varepsilon}\ge r\alpha(u)^{1-\varepsilon}, \end{align*} $$

and this completes the proof.

3.3 Optimality of the exponent

We now prove the claimed optimality of Theorem 3.1.

Proposition 3.3. Let notation be as in Theorem 3.1. Then there is a constant $R>0$ such that for all $u=(u_1,\ldots ,u_n)\in ({\mathcal O}(K)^\times )^n$ , there exist $a_1,\ldots ,a_n\in {\mathcal O}(K)_{h}$ not all $0$ such that

$$ \begin{align*} a_1u_1+\cdots+a_n u_n=0 \end{align*} $$

and $h\le R\alpha (u)$ .

Proof. We assume, as we may, that $\alpha (u)=H(u)^{1/(n-1)}$ , for otherwise, we can omit some coordinates from u so that the identity holds.

By Dirichlet’s unit theorem, there exists $\unicode{x3bb} \in {\mathcal O}(K)^\times $ such that

$$ \begin{align*} \max(|\unicode{x3bb} u_1|_v,\ldots,|\unicode{x3bb} u_n|_v)\le C_{0}H(u) \end{align*} $$

for all $v\in M_{K,\infty }$ and some $C_{0}>0$ depending only on K. Up to replacing u by $\unicode{x3bb} u$ , which does not affect the height, we may assume $\unicode{x3bb} =1$ . It follows that for all $a_1,\ldots ,a_n\in {\mathcal O}(K)_{h}$ ,

$$ \begin{align*} a_1u_1+\cdots+a_n u_n\in {\mathcal O}(K)_{C_{1}hH(u)}, \end{align*} $$

where $C_{1}=nC_{0}$ .

We observe that

$$ \begin{align*} c_2 h^{[K:\mathbb{Q}]}\le |{\mathcal O}(K)_h|\le C_2 h^{[K:\mathbb{Q}]} \end{align*} $$

for some constants $c_2,C_2>0$ depending only on K provided h is sufficiently large. This means that there are at least $c_2^n (h/2)^{n[K:\mathbb {Q}]}$ choices for $a_1,\ldots ,a_n\in {\mathcal O}(K)_{h/2}$ and there are at most $C_2(C_{1}hH(u))^{[K:\mathbb {Q}]}$ possible values for $a_1 u_1+\cdots +a_n u_n$ .

Now we take $h=R \,H(u)^{1/(n-1)}$ for a suitably large constant R so that

$$ \begin{align*} c_{2}^n (h/2)^{n[K:\mathbb{Q}]}>C_2(C_{1}hH(u))^{[K:\mathbb{Q}]}. \end{align*} $$

Dirichlet’s box principle implies that there are $b, \widetilde b \in ({\mathcal O}(K)_{h/2})^n$ such that $b\neq \widetilde b$ and

$$ \begin{align*} b_1u_1+\cdots+b_n u_n=\widetilde b_1 u_1+\cdots+\widetilde b_n u_n. \end{align*} $$

The claim follows by taking $a_j=b_j-\widetilde b_j$ .

4 Growth of eigencharacters

Let $\alpha :\mathbb {Z}^l \rightarrow \operatorname {\mathrm {Aut}}(\mathscr {N})$ be a morphism from $\mathbb {Z}^l$ to the group of automorphisms of a compact nilmanifold $\mathscr {N}=G/\Lambda $ . We assume that $\alpha (z)$ acts ergodically on $\mathscr {N}$ for every $z\neq 0$ and estimate the growth of the eigencharacters of $\alpha $ acting on the abelianized Lie algebra of G.

As in §2, we set $\mathfrak {t}=\mathfrak {g}/[\mathfrak {g},\mathfrak {g}]$ and identify $\mathfrak {t}$ with some $\mathbb {R}^d$ so that the projection $Z_{\Lambda }$ of $\log \Lambda $ in $\mathfrak {t}$ corresponds to $\mathbb {Z}^d$ . The representation $\alpha $ induces by differentiation a representation of $\mathbb {Z}^l$ on $\mathfrak {t}$ which preserves $Z_{\Lambda }$ . We call it $d_{\mathfrak {t}}\alpha : \mathbb {Z}^l\rightarrow \operatorname {\mathrm {GL}}^{\pm 1}_{d}(\mathbb {Z})$ , where $\operatorname {\mathrm {GL}}^{\pm 1}_{d}(\mathbb {Z})$ denotes the set of $d\times d$ -matrices with coefficients in $\mathbb {Z}$ and determinant $\pm 1$ . As $\mathbb {Z}^l$ is abelian, such a representation is triangularizable over $\overline {\mathbb {Q}}$ : we can decompose $\overline {\mathbb {Q}}^d$ as a direct sum of subrepresentations

$$ \begin{align*} \overline{\mathbb{Q}}^d=\oplus_{\chi \in \mathscr{X}}L_{\chi}, \end{align*} $$

where $\mathscr {X}$ is a (finite) set of characters $\chi : \mathbb {Z}^l \rightarrow \overline {\mathbb {Q}}^\times $ , and for each $\chi \in \mathscr {X}$ , the generalized eigenspace

$$ \begin{align*} L_{\chi}:=\{v\in \overline{\mathbb{Q}}^d | \text{ for all } z\in \mathbb{Z}^l, (d_{\mathfrak{t}}\alpha(z)-\chi(z) \text{Id})^d(v)=0 \} \end{align*} $$

is non-zero. Moreover, the Galois group $\mathcal {G}=\text {Gal}(\overline {\mathbb {Q}}/\mathbb {Q})$ acting coordinate-wise on $\overline {\mathbb {Q}}^d$ permutes these eigenspaces according to the formula $\sigma. L_{\chi }=L_{\sigma \circ \chi }$ .

The assumption of ergodicity on $\alpha $ can be reformulated as follows.

Lemma 4.1. (Exponential growth of eigencharacters)

There exists $c>0$ such that for every $\chi _{0} \in \mathscr {X}$ and for every $z\in \mathbb {Z}^l\smallsetminus \{0\}$ ,

$$ \begin{align*} \max_{\chi \in \mathcal{G}.\chi_{0}}|\chi (z)|\geq e^{c\lVert z\rVert}. \end{align*} $$

Proof. We recall the proof given in [Reference Gorodnik and Spatzier10, Lemma 2.1]. Let $\chi _{0}\in \mathscr {X}$ . We claim that the morphism of $\mathbb {Z}$ -modules

$$ \begin{align*} \psi : \mathbb{Z}^l\rightarrow \mathbb{R}^{|\mathcal{G}.\chi_{0}|}, \,z\mapsto (\log |\chi(z)|)_{\chi\in \mathcal{G}.\chi_{0}} \end{align*} $$

is injective with discrete image. Its $\mathbb {R}$ -linear extension $\overline {\psi } : \mathbb {R}^l\rightarrow \mathbb {R}^{|\mathcal {G}.\chi _{0}|}$ is thus injective, whence satisfies $\lVert \overline {\psi }(z)\rVert \geq c_{\chi _{0}}\lVert z\rVert $ for some $c_{\chi _{0}}>0$ and every $z\in \mathbb {Z}^l$ . The lemma follows by finiteness of $\mathscr {X}$ .

To prove the claim, we argue by contradiction assuming the existence of a sequence $(z_{i})\in (\mathbb {Z}^l)^{\mathbb {N}}$ of non-zero vectors such that $\psi (z_{i})\rightarrow 0$ . This means that $|\chi (z_{i})|\rightarrow 1$ for every $\chi \in \mathcal {G}.\chi _{0}$ . In particular, the associated minimal polynomials $P_{i}= \prod _{\chi \in \mathcal {G}.\chi _{0}}(X-\chi (z_{i}))\in \mathbb {Z}[X]$ have bounded coefficients, hence belong to a finite subset of $\mathbb {Z}[X]$ . Necessarily, the sequence of vectors $(\chi (z_{i}))_{\chi \in \mathcal {G}.\chi _{0}}$ has a constant subsequence, yielding some $z\in \mathbb {Z}^l\smallsetminus 0$ such that $|\chi (z)|=1$ for every $\chi \in \mathcal {G}.\chi _{0}$ . By Kronecker’s theorem (see e.g. [Reference Bombieri and Gubler2, Theorem 1.5.9]), $\chi _{0}(z)$ is a root of unity. This contradicts the ergodicity of the toral automorphism induced by $\alpha (z)$ on $G/[G,G]\Lambda $ , whence that of $\alpha (z)$ on $G/\Lambda $ .

5 Proof of exponential n-mixing

We now proceed to the proof of Theorem 1.1. Fix $n \geq 2$ , $\theta \in (0, 1]$ . Let $\underline {z}=(z_{1}, \ldots , z_{n}) \in (\mathbb {Z}^l)^n$ and consider some $\theta $ -Hölder functions $f_{1}, \ldots , f_{n}\in \mathcal {H}^{\theta }(\mathscr {N})$ as well as translation parameters $g_{1}, \ldots , g_{n}\in G$ . We begin with two observations.

• • The integral at study can be rewritten as

  $$ \begin{align*} \int_{\mathscr{N}}\prod_{i=1}^nf_{i}(g_{i}\alpha(z_{i})x)\,dm(x)=\int_{\mathscr{S}}f_{1}\otimes \cdots\otimes f_{n} \,dm_{\mathscr{S}}, \end{align*} $$
  where $\mathscr {S}$ is an affine rational submanifold of $\mathscr {N}^n$ , defined as the image of the embedding
  $$ \begin{align*} \mathscr{N}\rightarrow \mathscr{N}^n, \,x\mapsto (g_{1}\alpha(z_{1})x, \ldots, g_{n}\alpha(z_{n})x). \end{align*} $$

• • Equipping $\mathscr {N}^n$ with the product Riemannian metric on $\mathscr {N}$ , we have

  $$ \begin{align*} \lVert f_{1}\otimes \cdots\otimes f_{n}\rVert_{\mathcal{H}^{\theta}}\leq \lVert f_{1}\rVert_{\mathcal{H}^{\theta}}\cdots \lVert f_{n}\rVert_{\mathcal{H}^{\theta}}. \end{align*} $$

These observations reduce Theorem 1.1 to showing that the submanifold $\mathscr {S}$ is $\delta $ -equidistributed with respect to $\mathcal {H}^{\theta }(\mathscr {N}^n)$ for some $\delta <{C}/{N(\underline {z})^\eta }$ , where $N(\underline {z})=\exp (\min _{1\leq i\neq j\leq n}\lVert z_{i}-z_{j}\rVert )$ and $C, \eta>0$ are constants allowed to depend on the initial data $(G,\Lambda , \alpha , n, \theta )$ or possibly related structures, but not on $\underline {z}$ nor the $g_{i}$ .

Denote by $L>0$ the constant associated to the product nilmanifold $\mathscr {N}^n$ and $\theta $ as in Proposition 2.1. Let $\delta \in (0, 1/2)$ be such that $\mathscr {S}$ is not $\delta $ -equidistributed in $\mathscr {N}^n$ . By Proposition 2.1, there exists $q=(q_{1}, \ldots , q_{n})\in (\mathbb {Z}^d)^n$ such that

$$ \begin{align*} 0< \lVert q\rVert< \bigg(\frac{1}{\delta}\bigg)^L \quad\text{and}\quad \langle q, d_{\mathfrak{t}}\alpha(\underline{z})\mathfrak{t}\rangle=0, \end{align*} $$

where for $\omega \in \mathfrak {t}$ , we set $d_{\mathfrak {t}}\alpha (\underline {z})\omega =(d_{\mathfrak {t}}\alpha (z_{1})\omega , \ldots , d_{\mathfrak {t}}\alpha (z_{n})\omega )\in \mathfrak {t}^n$ , and recall that $d_{\mathfrak {t}}\alpha (z)$ is the projection of the differential of $\alpha (z)$ to $\mathfrak {t}$ . Taking $\overline {\mathbb {Q}}$ -linear combinations of the above equality, we get for every $\omega \in \overline {\mathbb {Q}}^d$ ,

(1) $$ \begin{align} \langle q, d_{\mathfrak{t}}\alpha(\underline{z})\omega\rangle=0. \end{align} $$

Let us choose once and for all a basis $\beta _{\chi }$ of each generalized eigenspace $L_{\chi }$ for $\chi \in \mathscr {X}$ whose vectors have algebraic integer coefficients and such that $d_{\mathfrak {t}}\alpha (\mathbb {Z}^l)$ is represented by upper-triangular matrices. As the $L_{\chi }$ span $\overline {\mathbb {Q}}^d$ and q is non-zero, there must exist $\chi _{0}\in \mathscr {X}$ such that $ \langle q_{i}, L_{\chi _{0}}\rangle \neq \{0\}$ for some $i\in \{1, \ldots ,n\}$ . We let $\omega _{0}$ be the first element of the basis $\beta _{\chi _{0}}$ such that $ \langle q_{i}, \omega _{0}\rangle \neq 0$ for some i. We can then write for every i,

$$ \begin{align*} \langle q_{i}, d_{\mathfrak{t}}\alpha(z_{i})\omega_{0}\rangle =\chi_{0}(z_{i}) \langle q_{i}, \omega_{0}\rangle \end{align*} $$

and equation (1) with $\omega =\omega _{0}$ yields

$$ \begin{align*} \sum_{i=1}^n\chi_{0}(z_{i}) \langle q_{i}, \omega_{0}\rangle=0. \end{align*} $$

We let K be the number field generated by the coefficients of the vectors belonging to the basis $(\beta _{\chi })_{\chi \in \mathscr {X}}$ . For every i, we then have $\langle q_{i}, \omega _{0}\rangle \in {\mathcal O}(K)$ and $\chi _{0}(z_{i})\in {\mathcal O}(K)^\times $ (eigenvalues of the matrix $d_{\mathfrak {t}}\alpha (z_{i})\in \operatorname {\mathrm {GL}}_{d}^{\pm 1}(\mathbb {Z})$ are algebraic units, and the relation $d_{\mathfrak {t}}\alpha (z_{i})\omega _{0}=\chi _{0}(z_{i})w_{0}$ forces $\chi _{0}(z_{i})$ to be in K). Fix $\varepsilon \in (0,1)$ arbitrarily. Theorem 3.1 yields a constant $r>0$ depending only on $(K, n, \varepsilon )$ such that, for some $i_{0}\in \{1, \ldots , n\}$ and some Galois automorphism $\sigma \in \mathcal {G}$ ,

(2) $$ \begin{align} |\langle q_{i_{0}}, \sigma(\omega_{0})\rangle|\geq r \alpha(u)^{1-\varepsilon}, \end{align} $$

where $u=(\chi _{0}(z_{1}), \ldots , \chi _{0}(z_{n}))$ . The exponential growth of eigencharacters presented in Lemma 4.1 yields a lower bound on $\alpha (u)$ :

$$ \begin{align*} \alpha(u)\,&\geq \min_{1\leq i\neq j\leq n}(H(\chi_{0}(z_{i}), \chi_{0}(z_{j})))^{1/(n-1)}\\ &=\min_{1\leq i\neq j\leq n} \prod_{v\in M_{K, \infty}} \max (|\chi_{0}(z_{i}-z_{j})|_{v}, 1)^{{1}/{((n-1)[K:\mathbb{Q}])}}\\ &\geq \min_{1\leq i\neq j\leq n} \exp{\bigg(\frac{c \lVert z_{i}-z_{j}\rVert}{(n-1)[K:\mathbb{Q}]}\bigg)}. \end{align*} $$

Plugging this lower bound into equation (2) and using the Cauchy–Schwarz inequality, we get

$$ \begin{align*} \lVert q\rVert\geq r c_{1} \exp{\bigg(\frac{c(1-\varepsilon)}{(n-1)[K:\mathbb{Q}]} \min_{1\leq i\neq j\leq n}\lVert z_{i}-z_{j}\rVert\bigg)}, \end{align*} $$

where $c_{1}=\min _{\sigma \in \mathcal {G}}\,\lVert \sigma (\omega _{0})\rVert ^{-1}$ only depends on our choice of $(\beta _{\chi })_{\chi \in \mathscr {X}}$ . Recalling that $\delta ^{-L}> \lVert q\rVert $ , we obtain that

$$ \begin{align*} \delta< r c_{1} N(\underline{z})^{-\eta}, \quad\text{where}\quad \eta:=\frac{c(1-\varepsilon)}{(n-1)[K:\mathbb{Q}]L}. \end{align*} $$

To sum up the above discussion, we have proven that for any $\underline {z}\in (\mathbb {Z}^l)^n$ satisfying $r c_{1} N(\underline {z})^{-\eta } <1/2$ , the submanifold $\mathscr {S}$ of $\mathscr {N}^n$ is $r c_{1} N(\underline {z})^{-\eta }$ -equidistributed for $\mathcal {H}^{\theta }(\mathscr {N}^n)$ . It follows that for all $\underline {z}\in (\mathbb {Z}^l)^n$ and all $f_{1}, \ldots , f_{n}\in \mathcal {H}^{\theta }(\mathscr {N})$ , $g_{1}, \ldots , g_{n}\in G$ , we have

$$ \begin{align*} \bigg|\int_{\mathscr{N}}\prod_{i=1}^n f_{i}(g_{i}\alpha(z_{i})x) \,dm(x) - \prod_{i=1}^n \int_{\mathscr{N}}f_{i}(x) \,dm(x)\bigg| \leq \frac{4rc_{1}}{ N(\underline{z})^{\eta}} \prod_{i=1}^n \lVert f_{i}\rVert_{\mathcal{H}^\theta} \end{align*} $$

and this concludes the proof. (The constant $4$ on the right is needed to get a valid statement when $r c_{1} N(\underline {z})^{-\eta } \ge 1/2$ .)