A set of 2-recurrence whose perfect squares do not form a set of measurable recurrence

1.1 Outline of the article

Our approach is similar to Kriz’s construction [Reference Kříž18] proving that there is a set of topological recurrence which is not a set of measurable recurrence. Very roughly, our example S in Theorem 1.1 is $\{n:n^2 \in R\}$ , where R is Kriz’s example. While this description is not quite correct, it may help those familiar with [Reference Kříž18], [Reference Griesmer16] or [Reference Griesmer15] understand our construction.

The overall proof of Theorem 1.1 is presented at the end of §2. We outline its components here. Section 2 begins by collecting standard facts about the following finite approximations to recurrence properties.

Lemma 2.1 says that if $S_1, S_2\subseteq \mathbb Z$ are finite, $\delta _1$ -non-recurrent, and $\delta _2$ -non-recurrent, then for all sufficiently large m, $S_1\cup mS_2$ is $2\delta _1\delta _2$ -non-recurrent. Thus, if $S_1^{\wedge 2}$ and $S_2^{\wedge 2}$ are $\delta _1$ -non-recurrent and $\delta _2$ -non-recurrent, respectively, then $(S_1 \cup mS_2)^{\wedge 2}$ is $2\delta _1\delta _2$ -non-recurrent for all sufficiently large m, as $(S_1 \cup mS_2)^{\wedge 2} = S_1^{\wedge 2} \cup m^2S_2^{\wedge 2}$ .

Lemma 2.3 says that $S\subseteq \mathbb Z$ is $\delta $ -non-recurrent if and only if for all $\delta '< \delta $ and all finite subsets $S'\subseteq S$ , $S'$ is $\delta '$ -non-recurrent. Likewise, if $S\subseteq \mathbb Z$ is $(\eta ,2)$ -recurrent, then for all $\eta '<\eta $ , there is a finite subset $S'\subseteq S$ which is $(\eta ',2)$ -recurrent.

The proof of Theorem 1.1 is given at the end of §2; it explains in detail how finite approximations are assembled to form a $2$ -recurrent set whose perfect squares do not form a set of measurable recurrence. This reduces the problem to proving Lemma 2.4, which states that the required finite approximations exist. These approximations are based on Bohr–Hamming balls, which we introduce in §3. Bohr–Hamming balls were used in [Reference Griesmer15, Reference Kříž18] to construct sets with prescribed recurrence properties. Fixing $\delta <\tfrac 12$ and $\eta>0$ , Lemmas 3.4 and 3.5 show that there is a Bohr–Hamming ball $BH$ which is $\delta $ -non-recurrent, while $\sqrt {BH}:=\{n\in \mathbb N: n^2\in BH\}$ is $(\eta ,2)$ -recurrent.

The proof of Lemma 3.5 occupies §§4–15. It is proved by estimating multiple ergodic averages of the form

(1.1) $$ \begin{align} \lim_{N\to\infty} \frac{1}{N}\sum_{n=1}^N g(n^2\boldsymbol{\beta}) \int f\cdot f\circ T^n \cdot f\circ T^{2n}\, d\mu, \end{align} $$

where $(X,\mathcal B,\mu ,T)$ is a measure-preserving system, $f:X\to [0,1]$ has $\int f\, d\mu>\delta $ for some prescribed $\delta>0$ , $\boldsymbol {\beta }\in \mathbb T^r$ for some $r\in \mathbb N$ , and $g:\mathbb T^r\to [0,1]$ is Riemann integrable. Under certain hypotheses on g, we will prove the limit in equation (1.1) is positive; this is the inequality in equation (4.7) in the proof of Lemma 3.5. In §4, we show how the general case may be reduced to that where T is totally ergodic. The remainder of the article, outlined in §5.2, is dedicated to analyzing the limit in equation (1.1) when T is totally ergodic. Section 8 shows that the totally ergodic case can be further reduced to the study of standard $2$ -step Weyl systems, and §§9–13 are dedicated to simplifying and estimating equation (1.1) for these systems.

Readers familiar with the theory of characteristic factors (especially [Reference Frantzikinakis6]) may find it most profitable to read §§2, 3, 5, and 8 in detail, and skim §4.

3.1 Cylinders and Fourier coefficients

Here we define constituents of approximate Hamming balls.

Definition 3.6. Given $r\in \mathbb N$ , $I\subseteq \{1,\ldots ,r\}$ , $\eta>0$ , and $\mathbf y\in \mathbb T^r$ , define the $\eta $ -cylinder determined by I around $\mathbf y$ to be

$$ \begin{align*} V_{I,\mathbf y,\eta}:=\{\mathbf x\in \mathbb T^r : \|x_i-y_i\|<\eta \text{ for all } i \in I\},\end{align*} $$

so that

(3.1) $$ \begin{align} U:=\operatorname{Hamm}(\mathbf y;k,\eta) = \bigcup_{\substack{I\subseteq \{1,\ldots, r\}\\ |I| = r-k}} V_{I,\mathbf y, \eta}. \end{align} $$

We say that $g:\mathbb T\to \mathbb R$ is a cylinder function subordinate to U if $g={m(V)}^{-1}1_V$ , where V is one of the cylinders $V_{I,\mathbf y,\eta }$ in equation (3.1). Note that each cylinder function subordinate to U is supported on U.

Let $\mathcal S^1$ denote the circle group $\{z\in \mathbb C:|z|=1\}$ with the usual topology and the group operation of complex multiplication. If Z is a compact abelian group with Haar probability measure m, $\widehat {Z}$ denotes its Pontryagin dual, meaning $\widehat {Z}$ is the group of continuous homomorphisms $\chi :Z\to \mathcal S^1$ ; such homomorphisms are called characters of Z. Given $f:Z\to \mathbb C$ , its Fourier transform is $\hat {f}:\widehat {Z}\to \mathbb C$ given by $\hat {f}(\chi )=\int f \overline {\chi }\, dm$ .

For $s\in Z$ , let $f_s$ be the translate of f defined by $f_s(x):=f(x+s)$ . Then $\widehat {f_s}(\chi )=\chi (s)\hat {f}(\chi )$ for each $\chi \in \widehat {Z}$ .

As usual, for $f, g: Z\to \mathbb C$ , $f*g$ denotes their convolution, defined as $f*g(x):=\int f(t)g(x-t)\, dm(t)$ . We will use the standard identity $\widehat {f*g}=\hat {f}\hat {g}$ (the Fourier transform turns convolution into pointwise multiplication).

Letting $\|f\|:=(\int |f|^2\, dm)^{1/2}$ denote the $L^2(m)$ norm of f, we have the standard Plancherel identity in equation (3.2), which leads to the subsequent lemma,

(3.2) $$ \begin{align} \sum_{\chi\in \widehat{Z}} |\hat{f}(\chi)|^2 = \|f\|^2. \end{align} $$

Proof. Let $S_1 = \{\chi _1,\ldots , \chi _k\}$ be the set of characters attaining the k largest values of $|\hat {f}|$ , let $S_2 = \widehat {Z}\setminus S_1$ , and let $c=\max \{|\hat {f}(\chi )|:\chi \in S_2\}$ . By definition, we have $|\hat {f}(\chi )|\geq c$ for all $\chi \in S_1$ .

We split the left-hand side of equation (3.2) into sums over $\chi \in S_1$ and $\chi \in S_2$ , then subtract the sum over $S_1$ to get

$$ \begin{align*}\sum_{\chi \in S_2} |\hat{f}(\chi)|^2 = \|f\|^2 - \sum_{\chi\in S_1} |\hat{f}(\chi)|^2.\end{align*} $$

Since $|\hat {f}(\chi )|\geq c$ for all $\chi \in S_1$ , the right-hand side is bounded above by $\|f\|^2 - kc^2$ . Since $c\leq |\hat {f}(\chi )|$ for at least one $\chi \in S_2$ , the left-hand side above is bounded below by $c^2$ . So

$$ \begin{align*} c^2\leq \sum_{\chi \in S_2} |\hat{f}(\chi)|^2 = \|f\|^2 - \sum_{\chi\in S_1} |\hat{f}(\chi)|^2 \leq 1-kc^2, \end{align*} $$

which implies $c^2\leq 1-kc^2$ . Solving, we get $c\leq (1+k)^{-1/2}$ . This means $|\hat {f}(\chi )|< k^{-1/2}$ for all $\chi \in S_2$ .

Much of the proof of Lemma 3.5 is contained in Lemma 3.9. The actual application requires a technical generalization (Lemma 12.2).

Proof. (i) Assuming k, r, $\chi _j$ , and U are as in the statement, we may write $\chi _j$ as

(3.3) $$ \begin{align} \chi_j(x_1,\ldots,x_r)=e\bigg( \sum_{l=1}^{r} n^{(j)}_lx_l\bigg), \end{align} $$

where $e(t):=\exp (2\pi i t)$ and $n^{(j)}_l\in \mathbb Z$ . Non-triviality of $\chi _j$ means that for each j, at least one of the $n^{(j)}_l$ is non-zero. So choose one such index $l_j$ for each $j\leq k$ and let $I=\{1,\ldots ,r\}\setminus \{l_1,\ldots ,l_k\}$ . In case some of the $l_j$ repeat, remove additional elements from I so that $|I|=r-k$ .

Writing U as $\operatorname {Hamm}(\mathbf y;k,\eta )$ , let $V = V_{I,\mathbf y,\eta }=\{\mathbf x\in \mathbb T^d:\|y_l-x_l\|<\eta \text { for all } l \in I\}$ , so that $V\subseteq U$ . Let $g:={m(V)}^{-1}1_V$ , so that g is a cylinder function subordinate to U, and let $j\leq k$ . To prove that $\hat {g}(\chi _j)=0$ , note that g does not depend on any of the coordinates $x_{l_j}$ , so we can simplify the right-hand side of equation (3.3) as $e(\sum _{\substack {l=1 \\ l\neq l_j}}^{r} n_{l}^{(j)}x_l)e(n_{l_j}^{(j)} x_{l_j})$ and write $\hat {g}(\chi _j)=\int g \overline {\chi }_j \, dm$ as

$$ \begin{align*} \int_{\mathbb T^{r-1}} g(x_1,\ldots,x_r) e\bigg(-\sum_{\substack{l=1 \\ l\neq l_j}}^{r} n_{l}^{(j)}x_l\bigg)\, dx_1\ldots \, dx_{l_{j-1}}\, dx_{l_{j+1}}\, \ldots \, dx_{r} \, \int_{\mathbb T} e(-n_{l_j}^{(j)} x_{l_j})\, dx_{l_j}. \end{align*} $$

Since $\int e(-n_{l_j}^{(j)} x_{l_j})\, dx_{l_j}=0$ , we conclude that $\hat {g}(\chi _j)=0$ for each j. To complete the proof of part (i), we observe that $\widehat {g_{s}}(\chi )=\chi (s)\hat {g}(\chi )$ for each $\chi $ .

To prove part (ii), assume $f:\mathbb T^r\to \mathbb C$ has $\|f\|\leq 1$ , and let $|\hat {f}(\chi _1)|, \ldots , |\hat {f}(\chi _k)|$ be the k largest values of $|\hat {f}|$ . By part (i), choose a cylinder function g subordinate to U so that $\hat {g}(\chi _j)=0$ for these $\chi _j$ . Then $|\hat {f}(\chi )|< k^{-1/2}$ for all other $\chi $ , by Lemma 3.7. Note that $|\hat {g}(\chi )|\leq 1$ for all $\chi \in \widehat {\mathbb T}^d$ , since $\int |g|\, dm =1$ . We therefore have $\widehat {f*g}(\chi _j)=\hat {f}(\chi _j)\hat {g}(\chi _j)=0$ for $j=1,\ldots ,k$ , while $|\widehat {f*g}(\chi )|\leq |\hat {f}(\chi )|<k^{-1/2}$ for all other $\chi $ .

4.1 Factors and extensions

If $\mathbf X = (X,\mathcal B,\mu ,T)$ and $\mathbf Y = (Y,\mathcal D,\nu ,S)$ are MPSs, we say that $\mathbf Y$ is a factor of $\mathbf X$ if there is a measurable $\pi : X\to Y$ intertwining S and T, meaning

$$ \begin{align*} \pi(Tx) = S\pi(x) \quad \text{for } \mu\text{-almost every (a.e.)} x\in X, \end{align*} $$

and $\mu (\pi ^{-1}D) = \nu (D)$ for all $D\in \mathcal D$ . Strictly speaking, the factor is the pair $(\pi , \mathbf Y)$ , and we refer to ‘the factor $\pi :\mathbf X\to \mathbf Y$ ’.

If $\pi :\mathbf X\to \mathbf Y$ is a factor and $f\in L^2(\mu )$ is equal $\mu $ -almost everywhere to a function of the form $g\circ \pi $ , with $g\in L^2(\nu )$ , we say that f is $\mathbf Y$ -measurable. This is equivalent to saying that f is $\pi ^{-1}(\mathcal D)$ -measurable (modulo $\mu $ ). We denote by $P_{\mathbf Y}$ the orthogonal projection from $L^2(\mu )$ to the space of $\pi ^{-1}(\mathcal D)$ -measurable functions. Given $f\in L^2(\mu )$ , we identify $P_{\mathbf Y}f$ with $\tilde {f}\in L^2(\nu )$ satisfying $P_{\mathbf Y} f = \tilde {f}\circ \pi $ .

We repeatedly use, without comment, the fact that $P_{\mathbf Y}$ is a positive operator preserving integration with respect to $\mu $ . In other words, if $f(x)\geq 0$ for $\mu $ -a.e. x, then $P_{\mathbf Y}f(x)\geq 0$ for $\mu $ -a.e. x, and $\int f\, d\mu = \int P_{\mathbf Y} f\, d\mu $ . Consequently, $\sup f \geq \tilde {f}(y)\geq \inf f$ for $\nu $ -a.e. y and $\int \tilde {f}\, d\nu = \int f\, d\mu $ .

4.2 Reducing to total ergodicity

The next lemma is used to deduce Lemma 3.5 from Lemma 4.4 and Theorem 4.1. Part (i) is a special case of [Reference Bergelson, Host and Kra2, Corollary 4.6], and part (ii) is an immediate consequence of part (i). Here ‘ $\mathbf Y$ is an inverse limit of ergodic nilsystems’ means that for all $f\in L^\infty (\nu )$ and $\varepsilon>0$ , there is a factor $\pi :\mathbf Y\to \mathbf Z$ , where $\mathbf Z=(Z,\mathcal Z,\eta ,R)$ is an ergodic nilsystem and $\|f-P_{\mathbf Z}f\|_{L^1(\nu )}<\varepsilon $ .

To derive part (ii) from part (i), note that

$$ \begin{align*} &\frac{1}{N}\sum_{n=1}^N \bigg|\int f\cdot T^{b_n}f \cdot T^{2b_n}f \, d\mu - \int \tilde{f}\cdot S^{b_n}\tilde{f} \cdot S^{2b_n}\tilde{f}\, d\nu \bigg| \\ &\quad\leq \frac{b_N}{N} \cdot \frac{1}{b_N}\sum_{n=1}^{b_N}\bigg| \int f_0\cdot T^n f_1\cdot T^{2n}f_2 \, d\mu - \int \tilde{f}_0\cdot S^n \tilde{f}_1\cdot S^{2n}\tilde f_2\, d\nu\bigg|\\ &\quad\underset{N\to\infty}{\longrightarrow} 0, \end{align*} $$

since ${b_N}/{N}$ is bounded.

We get the next result by combining the definition of ‘inverse limit’ with the fact that for every ergodic nilsystem $(Y,\mathcal D,\nu ,S)$ , there is an $\ell \in \mathbb N$ such that the ergodic components of $(Y,\mathcal D,\nu ,S^\ell )$ are totally ergodic; see [Reference Frantzikinakis6, Proposition 2.1] for justification.

Remark 4.10. Here we identify a technical difficulty common in multiple recurrence arguments. Readers familiar with the use of Markov’s inequality to overcome this difficulty may skip to the proof of Lemma 3.5.

Our proof of Lemma 3.5 starts with an ergodic, but not totally ergodic, MPS $\mathbf X=(X,\mathcal B,\mu ,T)$ . By Lemma 4.7, it suffices to prove the lemma in the special case where $\mathbf X$ is an inverse limit of ergodic nilsystems, so we assume $\mathbf X$ is such an inverse limit. We then consider $f:X\to [0,1]$ with $\int f\, d\mu>\delta $ . The goal is to find an $\ell \in \mathbb N$ and a sequence $(b_n)$ of elements of $\sqrt {BH}/\ell $ satisfying equation (4.7). The main difficulty arises when trying to exploit the structure of nilsystems: Lemma 4.4 requires total ergodicity, so we fix $\varepsilon>0$ and choose a factor $\pi :\mathbf X\to \mathbf Y$ where $\mathbf Y$ is an ergodic nilsystem satisfying parts (i) and (ii) in Lemma 4.8. We choose $\ell $ so that the ergodic components of $(Y,\mathcal D,\nu ,S^\ell )$ are totally ergodic, and we enumerate these components as $\mathbf Y_i = (Y_i, \mathcal D_i, \nu _i,S^\ell )$ , ${i=1,\ldots ,M}$ . With Notation 4.9 defined above, let $\tilde {f}\circ \pi =P_{\mathbf Y}f$ and $\tilde {f}_i=\tilde {f}|_{Y_i}$ . Lemma 4.4 allows us to choose, for each ergodic component $\mathbf Y_i$ where $\int \tilde {f}_i\, d\nu _i>\delta /2$ , a sequence $b_n^{(i)}\in \sqrt {BH}/\ell $ having linear growth, such that

(4.4) $$ \begin{align}\lim_{N\to\infty} \frac{1}{N}\sum_{n=1}^N \int \tilde{f}_i\cdot S^{\ell b_n^{(i)}} \tilde{f}_i \cdot S^{2\ell b_n^{(i)}}\tilde{f}_i\, d\nu_i>c(\delta/2)/2. \end{align} $$

The choice of $b_{n}^{(i)}$ depends on $\mathbf Y_i$ , so equation (4.4) implies only that

(4.5) $$ \begin{align} \lim_{N\to\infty} \frac{1}{N}\sum_{n=1}^N \int \tilde{f}\cdot S^{\ell b_n^{(i)}} \tilde{f} \cdot S^{2\ell b_n^{(i)}}\tilde{f}\, d\nu> \frac{1}{M}\frac{c(\delta/2)}{2}. \end{align} $$

If M is large, then $\|f-P_{\mathbf Y}f\|_{L^1(\mu )}$ may be large compared with $({1}/{M}){c(\delta /2)}/{2}$ , and equation (4.5) will not immediately imply equation (4.7). To overcome this obstacle, we want to find an i where equation (4.4) holds and $({1}/{M})\|f_i-P_{\mathbf Y}f_i\|_{L^1(\mu )}$ is sufficiently small to make $\int \tilde {f}_i\cdot S^{\ell a}\tilde {f}_i \cdot S^{\ell b}\tilde {f}_i\, d\nu _i$ close to $\int f_i\cdot T^{\ell a}f_i\cdot T^{\ell b} f_i\, d\mu _i$ for all $a, b$ . Such an i is provided by two straightforward applications of Markov’s inequality outlined in §15.3.

Before proving Lemma 3.5, we recall its statement: for all $\delta>0$ , there is a $k_0\in \mathbb N$ such that for every proper Bohr–Hamming ball $BH:=BH(\boldsymbol {\beta },\boldsymbol {y}; k, \eta )$ with $k\geq k_0$ , $\eta>0$ , and $\mathbf y\in \mathbb T^r$ , $\sqrt {BH}$ is $(\delta ,2)$ -recurrent.

Proof of Lemma 3.5, assuming Lemma 4.4

Let $\delta>0$ and choose $k_0\in \mathbb N$ so that for all $k\geq k_0$ , the inequality in equation (4.3) holds in Lemma 4.4 with $c(\delta /2)$ in place of $c(\delta )$ . Let $BH$ be a proper Bohr–Hamming ball with radius $(k,\eta )$ for some $\eta>0$ . It suffices to prove that for every MPS $(X,\mathcal B,\mu ,T)$ with $A\subseteq X$ having $\mu (A)>\delta $ ,

(4.6) $$ \begin{align} \mu(A\cap T^{-n}A\cap T^{-2n}A)> 0 \quad \text{for some } n\in \sqrt{BH}. \end{align} $$

By Remark 4.3, we need only consider ergodic MPSs. We will prove that if $\mathbf X$ is ergodic and $f:X\to [0,1]$ has $\int f\, d\mu>\delta $ , then there is a sequence of elements $b_n\in \sqrt {BH}$ with linear growth such that

(4.7) $$ \begin{align} \liminf_{N\to\infty} \frac{1}{N}\sum_{n=1}^N \int f\cdot T^{b_n}f\cdot T^{2b_n}f\, d\mu>0. \end{align} $$

The special case of equation (4.7) where $f=1_A$ implies equation (4.6), as the integral then simplifies to $\mu (A\cap T^{-b_n}A\cap T^{-2b_n}A)$ .

By part (ii) of Lemma 4.7, it suffices to prove equation (4.7) when $\mathbf X=(X,\mathcal B,\mu ,T)$ is an inverse limit of ergodic nilsystems. We now fix such an $\mathbf X$ , and $f:X\to [0,1]$ with $\int f\, d\mu>\delta $ .

Let $\varepsilon = ({\delta }/{24})c({\delta }/{2})$ , and let $\pi :\mathbf X\to \mathbf Y$ be the factor provided by Lemma 4.8 for this $\varepsilon $ , with $\ell \in \mathbb N$ chosen so that the ergodic components $\mathbf Y_i$ of $(Y,\mathcal D,\nu ,S^{\ell })$ are totally ergodic. Let M be the number of ergodic components (we can take $\ell = M$ , but we do not need this fact) so that $\mu (Y_i)=1/M$ for each i.

Let $X_i=\pi ^{-1}(Y_i)$ and let $f_i=1_{X_i}f$ , so that the $X_i$ partition X into sets of measure $1/M$ , and $\sum _i \int f_i \, d\mu = \int f \, d\mu>\delta $ . Observe that $P_{\mathbf Y}f_i$ is supported on $X_i$ and $\int P_{\mathbf Y}f_i\, d\mu = \int f_i\, d\mu $ for each i.

Setting $\mathbf Y_{i}:=(Y_i,\mathcal D_i,\nu _i,S^{\ell })$ , where $\nu _i:=M\nu |_{Y_i}$ , we get that $\mathbf{Y}_{i}$ is a totally ergodic MPS. Likewise, $\mathbf X_{i}:= (X_i, \mathcal B_i, \mu _i,T^{\ell })$ , with $\mu _i:=M\mu |_{X_i}$ is an MPS (possibly not ergodic), with $\pi |_{X_i}:X_i\to Y_i$ a factor map. To prove equation (4.7), we will find a sequence $b_n$ of elements of $\sqrt {BH}/\ell $ having linear growth and $i\leq M$ with

(4.8) $$ \begin{align} \liminf_{N\to\infty} \frac{1}{N}\sum_{n=1}^N \int f_i\cdot T^{\ell b_n}f_i\cdot T^{2\ell b_n} f_i\, d\mu> 0. \end{align} $$

We claim that there is an i such that

(4.9) $$ \begin{align} \int f_i\, d\mu &> \frac{\delta}{2M} \quad\text{and } \end{align} $$

(4.10) $$ \begin{align} \|f_i - P_{\mathbf Y}f_i\|_{L^1(\mu)}&<\frac{c(\delta/2)}{12M}. \end{align} $$

This i is provided by Lemmas 15.5 and 15.6: setting

$$ \begin{align*}I:=\bigg\{i:\int f_i\, d\mu>\frac{\delta}{2M}\bigg\}, \quad J:=\bigg\{i: \int |f_i - P_{\mathbf Y}f_i|\, d\mu < \frac{c(\delta/2)}{12M}\bigg\},\end{align*} $$

we get $|I|> M\delta /2$ and $|J| > M(1-12\varepsilon/c(\delta/2)) = M(1-\delta/2)$ . Thus $|I|+|J|>M$ , implying $I\cap J$ is non-empty.

Fix i satisfying inequalities (4.9) and (4.10). Note that inequality (4.9) and the definition of $\nu _i$ , $\mu _i$ , and $\tilde {f}_i$ imply

(4.11) $$ \begin{align} \int \tilde{f}_i\, d\nu_i> \delta/2. \end{align} $$

Since $(Y_i,\mathcal B_i,\nu _i,S^{\ell })$ is totally ergodic, we may apply Lemma 4.4 to choose a sequence of elements $b_n\in \sqrt {BH}/\ell $ having linear growth and satisfying

(4.12) $$ \begin{align} \lim_{N\to\infty} \frac{1}{N}\sum_{n=1}^N \int \tilde{f}_i\cdot S^{\ell b_n}\tilde{f}_i \cdot S^{2\ell b_n} \tilde{f}_i\, d\nu_i> c(\delta/2)/2. \end{align} $$

Inequality (4.10), the bounds $\|f_i\|_{\infty }\leq 1$ , $\|P_{\mathbf Y}f_{i}\|_{\infty }\leq 1$ , and Lemma 15.7 imply

(4.13) $$ \begin{align} \bigg|\int f_i\cdot T^{\ell a} f_i \cdot T^{\ell b} f_i\, d\mu_i - \int P_{\mathbf Y_i}f_i\cdot T^{\ell a} P_{\mathbf Y_i}f_i \cdot T^{\ell b} P_{\mathbf Y_i}f_i\, d\mu_i\bigg|< \frac{1}{4}c(\delta/2) \end{align} $$

for each $a,b\in \mathbb N$ . Recalling the definition of $\mu _i$ and $\nu _i$ , we see that for all sufficiently large N,

$$ \begin{align*} &\frac{1}{N} \sum_{n=1}^{N} \int f_i\cdot T^{\ell b_n} f_i \cdot T^{2\ell b_n} f_i\, d\mu\\ &\quad> \frac{1}{N}\sum_{n=1}^{N} \int \tilde{f}_i \cdot S^{\ell b_n} f_i \cdot S^{2\ell b_n} f_i\, d\nu - \frac{c(\delta/2)}{4M} \quad \text{by inequality}\ ({4.13})\\ &\quad> \frac{c(\delta/2)}{2M} - \frac{c(\delta/2)}{4M} \qquad\qquad\qquad\qquad\qquad\quad \ \text{ by inequality}\ ({4.12}) \\ &\quad= \frac{c(\delta/2)}{4M}. \end{align*} $$

The above inequalities imply equation (4.8). Since $f\geq f_i$ pointwise and we chose $b_n\in \sqrt {BH}/\ell $ , this implies equation (4.7) and completes the proof of Lemma 3.5.

5.1 Reformulation

Lemma 4.4 is an immediate consequence of the following reformulation. This version allows us to apply the theory of characteristic factors.

Lemma 5.1. Let $k<r\in \mathbb N$ , $\ell \in \mathbb N$ , let $\boldsymbol {\beta }\in \mathbb T^r$ be generating, and let $U\subseteq \mathbb T^r$ be an approximate Hamming ball of radius $(k,\eta )$ for some $\eta>0$ . For every totally ergodic MPS $(X,\mathcal B,\mu ,T)$ , and every measurable $f:X\to [0,1]$ , there is a cylinder function $g={m(V)}^{-1}1_V$ subordinate to U such that

(5.1) $$ \begin{align} \lim_{N\to \infty} \bigg|\frac{1}{N}\sum_{n=1}^N g(n^2\ell^2\boldsymbol{\beta})\int f\cdot T^{n} f \cdot T^{2n} f\, d\mu - L_3(f,T)\bigg|<2k^{-1/2}\|f\|^2. \end{align} $$

While U does not depend on f in Lemma 5.1, the choice of g to satisfy equation (5.1) does depend on f.

We prove Lemma 5.1 in §14. The derivation of Lemma 4.4 from Lemma 5.1 is an instance of the following general principle: if $a_n$ is a bounded sequence, $B\subseteq \mathbb N$ is enumerated as $\{b_1<b_2<\ldots \}$ , and $d(B):=\lim _{N\to \infty } ({|B\cap \{1,\ldots ,N\}|}/{N})>0$ , then

$$ \begin{align*} \lim_{N\to\infty} \frac{1}{N}\sum_{n=1}^N a_{b_n} = \lim_{N\to\infty} \frac{1}{Nd(B)}\sum_{n=1}^N1_{B}(n)a_n \end{align*} $$

provided the limit on the right exists. Note that $(b_n)_{n\in \mathbb N}$ has linear growth if $d(B)>0$ .

We will apply this principle with $a_n = \int f\cdot T^{n} f \cdot T^{2n} f\, d\mu $ and ${B=\{n:n^2\ell ^2\boldsymbol {\beta }\in V\}}$ , where V is a cylinder contained in U. Then, $g={m(V)}^{-1}1_V$ is a cylinder function subordinate to U, and $g(n^2\ell ^2\boldsymbol {\beta })={d(B)}^{-1}1_B(n)$ . The equation $d(B)=m(V)$ follows from Weyl’s theorem on uniform distribution (cf. §10). Note that this B is contained in $\sqrt {BH}/\ell $ , where $BH$ is the Bohr–Hamming ball corresponding to U, with frequency $\boldsymbol {\beta }$ .

5.2 Outline of a special case of Lemma 15.1

This outline highlights the key steps in our proof while avoiding some complications.

We begin with an arbitrary totally ergodic measure-preserving system $\mathbf X=(X,\mathcal B,\mu ,T)$ , $f: X\to [0,1]$ , and $k\in \mathbb N$ . We let $r>k$ , $\eta>0$ , and fix an approximate Hamming ball $U=\operatorname {Hamm}(\mathbf y;k,\eta )\subseteq \mathbb T^r$ and a generator $\boldsymbol {\beta } \in \mathbb T^r$ . We want to find a cylinder function g subordinate to U so that

(5.2) $$ \begin{align} A_N(f,g):= \frac{1}{N}\sum_{n=1}^N g(n^2\boldsymbol{\beta})\int f\cdot T^nf\cdot T^{2n}f\, d\mu \end{align} $$

satisfies $\lim _{N\to \infty } |A_N(f,g)-L_3(f,T)|<2k^{-1/2}\|f\|^2$ .

In §§6–8, we will reduce to the case where $\mathbf X$ is a standard 2-step Weyl system. This means that $(X,\mathcal B,\mu ,T)$ can be realized with $X=\mathbb T^d\times \mathbb T^d$ , $d\in \mathbb N$ , $\mu =$ Haar probability measure on $\mathbb T^d\times \mathbb T^d$ , and T is given by $T(x, y)=( x+\boldsymbol {\alpha },y+ x)$ , for some generator $\boldsymbol {\alpha }\in \mathbb T^d$ . The orbits of T can be computed explicitly: $T^n(x,y)=(x+n\boldsymbol {\alpha }, y+nx+\tbinom {n}{2}\boldsymbol {\alpha })$ . This reduction relies on the theory of characteristic factors, especially [Reference Frantzikinakis6, Theorem B].

To simplify this outline, we assume $r=d$ and $\boldsymbol {\beta } = \boldsymbol {\alpha }$ . We write functions on $\mathbb T^d\times \mathbb T^d$ with variables displayed as $f(x,y)$ , where $x, y\in \mathbb T^d$ . Writing $m\times m$ for Haar probability measure on $\mathbb T^d\times \mathbb T^d$ , we write $\int f\, dm\times m$ as $\int f(x,y)\, dx\, dy$ , or $\int f\binom {x}{y}\, dx\, dy$ to save space. With these assumptions, the averages in equation (5.2) become

$$ \begin{align*} B_N := \frac{1}{N}\sum_{n=1}^N g(n^2\boldsymbol{\alpha}) \int_{\mathbb T^d\times \mathbb T^d} f\,\Big(\begin{matrix} x\\[-2pt]y \end{matrix}\Big) \, f \begin{pmatrix} x+n\boldsymbol{\alpha}\\[-1pt]y+nx+\tbinom{n}{2}\boldsymbol{\alpha} \end{pmatrix} f\begin{pmatrix} x+2n\boldsymbol{\alpha} \\[-1pt] y+2nx + \tbinom{2n}{2}\boldsymbol{\alpha} \end{pmatrix} \, dx\, dy. \end{align*} $$

Proposition 13.1 provides an explicit formula for $\lim _{N\to \infty } B_N$ . Under the present assumptions, it says

(5.3) $$ \begin{align} &\lim_{N\to\infty} B_N\nonumber\\ &\quad= \int_{(\mathbb T^d)^4} f(x,y)f(x+s,y+t) \bigg(\int_{\mathbb T^d} f(x+2s,y+2t+2w)g(w) \, dw\bigg) \, ds\, dt \, dx\, dy. \end{align} $$

Write I for the right-hand side above, and define

$$ \begin{align*} f*_2 g(x,y):= \int f(x,y+2w) g(w)\, dw. \end{align*} $$

Using Lemma 12.2 (a generalization of Lemma 3.9), we choose a cylinder function g subordinate to U such that $|\widehat {f*_2 g}(\chi ,\psi )|<k^{-1/2}$ for all $(\chi ,\psi )\in \widehat {\mathbb T}^d\times \widehat {\mathbb T}^d$ with $\psi $ non-trivial. We set $f'(x)\kern1.2pt{:=}\kern1.2pt\int f(x,y)\, dy$ and $J'\kern1.2pt{:=}\kern1.2pt \int \int f'(x)f'(x\kern1.2pt{+}\kern1.2pt s)f'(x\kern1.2pt{+}\kern1.2pt 2s)\, dx\, ds$ . By Lemma 11.1, the bound on $\widehat {f*_2 g}$ will imply

(5.4) $$ \begin{align} |I - J'| < k^{-1/2}\|f\|^2. \end{align} $$

We can also prove (directly, or using Theorem 7.1) that $L_3(f,T)=J'$ . Combining equation (5.4) with equation (5.3), we then have equation (5.1), completing the outline of this special case. The factor $2$ on the right-hand side of equation (5.1) accounts for the reduction to Weyl systems.

In the general case, we must compute $\lim _{N\to \infty } A_N(f,g)$ for $d\neq r$ and $\boldsymbol {\beta }\neq \boldsymbol {\alpha }$ . The integral $\int f(x+2s,y+2t +2w) g(w)\, dw$ in equation (5.3) will then be replaced by an integral over an affine joining of $\mathbb T^d$ with $\mathbb T^r$ (Definition 9.4), but the computation in this case is not substantially different from the outline above.

5.3 Iterated integral notation

When all variables are displayed and there is no chance of confusion, we may omit all but one of the integral signs and the subscripts indicating the domain of integration. So the integral in equation (5.3) may be written as

$$ \begin{align*} \int f(x,y)f(x+s,y+t) f(x+2s,y+2t+2w)g(w) \, dw\, ds \, dt\, dx\, dy. \end{align*} $$

7.1 Kronecker factor of a standard 2-step Weyl system

A standard $2$ -step Weyl system is an MPS of the form $\mathbf Y = (Y, \mathcal B, m,S)$ , where $Y=\mathbb T^d\times \mathbb T^d$ , $d\in \mathbb N$ , and $S:Y\to Y$ is defined as $S(x,y)=(x+\alpha ,y+x)$ , for some fixed $\alpha =(\alpha _1,\ldots ,\alpha _d)$ generating $\mathbb T^d$ . There is an explicit formula for the orbits of S:

(7.2) $$ \begin{align} S^n(x,y)=(x+n\alpha, y + nx + \tbinom{n}{2}\alpha), \end{align} $$

which may be verified by induction. Ergodicity of $\mathbf Y$ is equivalent to $\mathbf \alpha $ generating $\mathbb T^d$ . For $d=1$ , this follows from [Reference Furstenberg12, Proposition 3.11, p. 67], and the general case follows from a nearly identical proof. Also explained in [Reference Furstenberg12] is the Kronecker factor of $\mathbf Y$ : the eigenfunctions of $\mathbf Y$ are exactly the functions $\chi $ on Y defined by

$$ \begin{align*} \chi((x_1,\ldots,x_d),(y_1,\ldots,y_d)):=\exp(2\pi i (n_1x_1+\cdots + n_dx_d)) \end{align*} $$

for some $n_j\in \mathbb Z$ , so the group of eigenvalues of $\mathbf Y$ is $\{\exp (2\pi i (n_1\alpha _1+\cdots +n_d\alpha _d)) : n_j\in ~\mathbb Z\}$ . Thus, the Kronecker factor of $\mathbf Y$ is obtained by setting $Z=\mathbb T^d$ and letting $\pi :\mathbb T^d\times \mathbb T^d \to \mathbb T^d$ be a projection onto the first coordinate. Since the span of the eigenfunctions of $\mathbf Y$ consists solely of those functions depending on the first coordinate, the orthogonal projection $P_{\mathbf Z}f(x,y)$ can be written as $(P_{\mathbf Z}f)(x,y):=\int f(x,y)\, dy$ . Combining this with Theorem 7.1, we have the following observation.

Observation 7.2. The Kronecker factor $(Z,\mathcal Z,m,R)$ of a standard 2-step Weyl system $(\mathbb T^d \times \mathbb T^d,\mathcal D,\mu ,S)$ is spanned by functions of the form $f(x,y)=g(x)$ (i.e. functions depending on only the first coordinate), and for all bounded $f:\mathbb T^d\times \mathbb T^d\to \mathbb C$ , we have

$$ \begin{align*} \lim_{N\to\infty} \frac{1}{N}\sum_{n=1}^N \int f\cdot S^n f\cdot S^{2n}f \, d\mu = \int_{\mathbb T^d}\int_{\mathbb T^d} f'(x)f'(x+s)f'(x+2s)\, dx\, ds, \end{align*} $$

where $f':\mathbb T^d\to \mathbb C$ is defined as $f'(x):=\int f(x,y)\, dy$ .