##  6\. Proof of Theorem 1.14

In this section, we prove Theorem 1.14, restated here for the convenience of
the reader:

######  Theorem 6.1 (Theorem 1.14).

Let $G=\bigoplus_{n=1}^{\infty}{\mathbb{Z}}$. Let $l\in\mathbb{N}$. There
exists $P=P(l)$ such that, for any $a,b\in\mathbb{N}$ with $p\mid\gcd(a,b)$
for some prime $p\geq P$, there is an ergodic $G$-system
$\left(X,\mathcal{X},\mu,(T_{g})_{g\in G}\right)$ and a set $A\in\mathcal{X}$
with $\mu(A)>0$ such that

$\displaystyle\mu(A\cap T_{ag}^{-1}A\cap T_{bg}^{-1}A)\leq\mu(A)^{l}$

for every $g\neq 0$.

Rather than constructing a $\bigoplus_{n=1}^{\infty}{\mathbb{Z}}$-system
directly, we will instead construct a
$\bigoplus_{n=1}^{\infty}{\mathbb{Z}/p^{2}\mathbb{Z}}$-system. Since
$\bigoplus_{n=1}^{\infty}{\mathbb{Z}/p^{2}\mathbb{Z}}$ is a quotient of
$\bigoplus_{n=1}^{\infty}{\mathbb{Z}}$, the system we construct can be lifted
to an ergodic $\bigoplus_{n=1}^{\infty}{\mathbb{Z}}$-system. Hence, Theorem
1.14 follows from:

######  Theorem 6.2.

For any $a,b,l\in\mathbb{N}$, there exists a prime $p$ (sufficiently large),
an ergodic $\bigoplus_{n=1}^{\infty}\mathbb{Z}/p^{2}\mathbb{Z}$-system
$\textbf{X}=\left(X,\mathcal{X},\mu,(T_{g})_{g\in\bigoplus_{n=1}^{\infty}{\mathbb{Z}/p^{2}\mathbb{Z}}}\right)$,
and a set $A\in\mathcal{X}$ with $\mu(A)>0$ such that

$\displaystyle\mu(A\cap T_{pag}^{-1}A\cap T_{pbg}^{-1}A)\leq\mu(A)^{l}$

for every $g\neq 0$.

The proof of Theorem 6.2 is based on the following result of Behrend [Beh46].

######  Theorem 6.3.

Let $a,b\in\mathbb{N}$ be distinct and non-zero. There is an absolute constant
$c>0$ such that: for every $N\in\mathbb{N}$, there is a subset
$B\subseteq\\{0,1,...,N-1\\}$ such that $|B|>N\cdot e^{-c\sqrt{\log(N)}}$ and
$B$ contains no configurations of the form $\\{n,n+am,n+bm\\}$ for $m\not=0$.

For every prime number $p$, let $C_{p}=\\{z\in\mathbb{C}:z^{p}=1\\}$ denote
the group of all roots of unity of order $p$ and let $\omega_{p}=e^{2\pi
i/{p}}$ be the first $p$-th root of unity in $\mathbb{C}$. The following is an
immediate corollary of Behrend’s theorem.

######  Lemma 6.4.

Let $a,b\in\mathbb{N}$ be distinct, then for every $l$, there exists a
sufficiently large prime $p$ and a subset $B\subseteq C_{p}$ of size
$|B|>p^{1-\frac{1}{l-1}}$ which contains no configurations of the form
$\\{y,y\cdot x^{a},y\cdot x^{b}\\}$ for $x\not=1$.

Throughout this section, we let $\mathcal{T}_{p}:=C_{p}^{\mathbb{N}}$ and
$G_{p}:=\bigoplus_{i\in I}\mathbb{Z}/p\mathbb{Z}$.

We start by giving a proof that the large intersection property fails for non-
ergodic systems.

######  Lemma 6.5.

Let $a,b\in\mathbb{Z}$ be distinct and nonzero. For every $L\in\mathbb{N}$,
there is a $P=P(L)$, such that for every prime $p\geq P$, there is a
$G_{p}$-system $(X,\mathcal{X},\mu,(T_{g})_{g\in G_{p}})$ such that, for every
$l\leq L$, there is a measurable set $A=A(l)$ with $\mu(A)>0$ and

$\mu(A\cap T_{ag}A\cap T_{bg}A)\leq\mu(A)^{l}$

for every $g\not=0$.

This result was previously established in [ABB21, Proposition 10.11], but we
give a different proof that will be useful later on.

###### Proof.

Let $p$ be a prime number and let $X_{p}=\mathcal{T}_{p}\times C_{p}$. We
equip $X_{p}$ with the Borel $\sigma$-algebra, the Haar measure $\mu$, and the
action of $G_{p}$ by

$T_{g}(x,u)=(x,\prod_{i=1}^{\infty}x_{i}^{g_{i}}u).$

Now fix a subset $B\subseteq C_{p}$ which avoids configurations of the form
$\\{y,y\cdot x^{a},y\cdot x^{b}\\}$ whenever $x\not=1$, and let
$A=\mathcal{T}_{p}\times B$. It is easy to see that $\mu(A)=\frac{|B|}{p}$ and
we have

$\displaystyle\mu(A\cap T_{ag}A\cap
T_{bg}A)=\int_{\mathcal{T}_{p}^{2}}1_{B}(y)1_{B}\left(y\prod_{i\in
I}x^{ag_{i}}\right)1_{B}\left(y\prod_{i\in I}x^{bg_{i}}\right)dxdy$
$\displaystyle=$
$\displaystyle\int_{\mathcal{T}_{p}^{2}}1_{B}(y)1_{B}\left(y\cdot\left(\prod_{\\{i~{}:~{}g_{i}\not=0\\}}x_{i}\right)^{a}\right)1_{B}\left(y\cdot\left(\prod_{\\{i~{}:~{}g_{i}\not=0\\}}x_{i}\right)^{b}\right)dxdy$
$\displaystyle=$
$\displaystyle\mu_{\mathcal{T}_{p}^{2}}\left(\left\\{(y,x)\in\mathcal{T}_{p}^{2}:\left\\{y,y\cdot\left(\prod_{\\{i~{}:~{}g_{i}\not=0\\}}x_{i}\right)^{a},y\cdot\left(\prod_{\\{i~{}:~{}g_{i}\not=0\\}}x_{i}\right)^{b}\right\\}\subset
B\right\\}\right).$

But,
$\left\\{y,y\cdot\left(\prod_{\\{i~{}:~{}g_{i}\not=0\\}}x_{i}\right)^{a},y\cdot\left(\prod_{\\{i~{}:~{}g_{i}\not=0\\}}x_{i}\right)^{b}\right\\}\subset
B$ if and only if $\prod_{\\{i~{}:~{}g_{i}\not=0\\}}x_{i}=1$. Since $g\not=0$,
we deduce that $\mu(A\cap T_{ag}A\cap
T_{bg}A)=\frac{|B|}{p^{2}}=\frac{p^{l-2}}{|B|^{l-1}}\mu(A)^{l}$. Now, choose
$P$ sufficiently large for which there exists a set $B$ with
$|B|>p^{1-\frac{1}{l-1}}$ (Lemma 6.4). Then $\mu(A\cap T_{ag}A\cap
T_{bg}A)<\mu(A)^{l}$ as required. ∎

Roughly speaking, the idea in this section is to construct an ergodic $p$-th
root for the system above.

We fix some $P$ sufficiently large as in Lemma 6.5, and let $p>P$ be a prime
number. For convenience of notations we let $\omega=e^{2\pi i/p}$ and
$\eta=e^{2\pi i/{p^{2}}}$. We define an action of
$G=\bigoplus_{n\in\mathbb{N}}\mathbb{Z}/p^{2}\mathbb{Z}$ on $\mathcal{T}$ by
setting $S_{g}x=\zeta(g)x$, where
$\zeta(g)=(\eta^{pg_{i}})_{i\in\mathbb{N}}=(\omega^{g_{i}})_{i\in\mathbb{N}}$.
Since the image of $\zeta$ is dense in $\mathcal{T}$, the action is ergodic.

Now, we extend this action to the product space $X=\mathcal{T}\times
C_{p^{2}}$. Let $\varphi:C_{p}\rightarrow C_{p^{2}}$ be the map

$\varphi(e^{\frac{2\pi ix}{p}})=e^{\frac{2\pi i|x|_{p}}{p^{2}}}$

where $|x|_{p}=x\mod p$. Then $\varphi$ is a cross-section of the canonical
projection $C_{p^{2}}\rightarrow C_{p}$ and we have that $\varphi(x)^{p}=x$,
and $\varphi(\omega)=\eta$. Our goal is to define an action $(T_{g})_{g\in G}$
on $X$ such that $T_{pg}(t,u)=(t,\prod_{i\in\mathbb{N}}t_{i}^{pg_{i}}\cdot
u)$.

We do so in two steps. We define an action $T^{\prime}_{g}$ on $X$ which
satisfies that $T^{\prime}_{e_{i}}(t,u)=(S_{e_{i}}t,\varphi(t_{i})u)$, for
every $i\in\mathbb{N}$, where
$e_{i}\in\bigoplus_{n=1}^{\infty}\mathbb{Z}/p^{n}\mathbb{Z}$ is the $i$-th
unit vector. Writing $g=\sum_{i\in\mathbb{N}}g_{i}e_{i}$ and using the group
law, we get the following action:

(6.1)
$T^{\prime}_{g}(t,u)=\left(S_{g}t,\prod_{j=1}^{\infty}\prod_{k=0}^{g_{j}-1}\varphi(\omega^{k}t_{j})\cdot
u\right)$

where an empty product $\prod_{k=0}^{-1}{x_{k}}$ is equal to 1.

Unfortunately, this action is not what we are looking for. Indeed,

$(T^{\prime}_{e_{j}})^{p}(t,u)=(t,\prod_{k=0}^{p-1}\varphi(\omega^{k}\cdot
t_{j})u)=(t,t_{j}\cdot\eta^{\binom{p}{2}}\cdot u).$

To fix that we let $\xi=\omega^{\frac{1-p}{2}}$ be a $p$-th root of
$\overline{\eta}^{\binom{p}{2}}$ and change the action accordingly:

(6.2)
$T_{g}(t,u)=\left(S_{g}t,\prod_{j=1}^{\infty}\left(\prod_{k=0}^{g_{j}-1}\varphi(\omega^{k}t_{j})\cdot\xi^{g_{j}}\right)\cdot
u\right).$

######  Lemma 6.6.

For every $t\in\mathcal{T}$, $u\in C_{p^{2}}$ and $g\in G$ we have

(6.3) $T_{pg}(t,u)=(t,t^{pg}u).$

###### Proof.

The proof is a direct computation. Indeed, it suffices to prove that (6.3)
holds for $g=e_{j}$ for every $j\in\mathbb{N}$. Let $j\in\mathbb{N}$ be
arbitrary. Since $\omega$ is of order $p$, $S_{pg}t=t$. As for the second
coordinate observe that

$\prod_{k=0}^{p-1}\varphi(\omega^{k}t_{j})\cdot\xi^{p}=\xi^{p}\cdot\eta^{\binom{p}{2}}\cdot
t_{j}=t_{j}.$

The first equality follows because the product is independent on $t_{j}$ and
always equals to
$\varphi(\omega)\cdot...\cdot\varphi(\omega^{p-1})=\eta^{\binom{p}{2}}$, and
the last equality follows from the definition of $\xi$. This completes the
proof of the lemma. ∎

The main difficulty in the proof is showing that this action is ergodic.

######  Lemma 6.7.

The action (6.2) on $X$ is ergodic.

###### Proof.

We use Zimmer criterion for ergodicity (Lemma 4.5). Since the action of $G$ on
$\mathcal{T}$ is ergodic, it is enough to show that the cocycle
$\sigma:G\times\mathcal{T}\rightarrow C_{p^{2}}$,
$\sigma(g,t)=\prod_{i=1}^{\infty}\prod_{k=0}^{g_{j}-1}\varphi(\omega^{k}t_{j})$
is minimal. Since $C_{p}$ is the largest proper subgroup of $C_{p^{2}}$, it is
enough to show that $\sigma$ is not cohomologous to a cocycle taking values in
$C_{p}$. Suppose by contradiction that there exists a cocycle
$\tau:\mathcal{T}\rightarrow C_{p}$ cohomologous to $\sigma$. Since
$\tau^{p}=1$, we deduce that
$\sigma(g,t)^{p}=\prod_{i=1}^{\infty}\omega^{\binom{g_{i}}{2}}t_{i}^{g_{i}}\xi^{pg_{i}}$
is a coboundary. Therefore, there exists $F:\mathcal{T}\rightarrow S^{1}$ such
that

(6.4) $\sigma^{p}(g,t)=\frac{F(S_{g}t)}{F(t)}$

for every $g\in G$ and $t\in T$. Observe that for every $g,h\in G$,
$\Delta_{h}\sigma^{p}(g,t)$ is a constant in $t$. Therefore, by (6.4),
$\Delta_{h_{1}}\Delta_{h_{2}}F$ is a constant for every $h_{1},h_{2}\in G$.
Let $s\in\mathcal{T}$ and define $\Delta_{s}F(x)=\frac{F(sx)}{F(x)}$. We claim
that $\Delta_{s}F(x)$ is an eigenfunction. Let $g_{1},g_{2}\in G$, then
$\Delta_{g_{1}}\Delta_{g_{2}}\Delta_{s}F(x)=\Delta_{s}\Delta_{g_{1}}\Delta_{g_{2}}F(x)=1$.
Hence, by ergodicity $\Delta_{g_{2}}\Delta_{s}F$ is constant and $\Delta_{s}F$
is an eigenfunction for every $s\in Z$. Recall that translations by $s\in Z$
are continuous with respect to the $L^{2}$-norm. In particular, there exists
an open subgroup $U\leq\mathcal{T}$ such that

(6.5) $\|\Delta_{s}F-1\|_{L^{2}(\mu_{\mathcal{T}})}<\sqrt{2}.$

By ergodicity, the multiplicity of each eigenvalue is $1$. Since
eigenfunctions with different eigenvalues are orthogonal, it follows that
$\Delta_{s}F$ is a constant for all $s\in U$. Otherwise, $\Delta_{s}F$ is
orthogonal to $1$ and then

$\|\Delta_{s}F-1\|_{L^{2}(\mu_{\mathcal{T}})}^{2}=\|\Delta_{s}F\|_{L^{2}}^{2}+\|1\|_{L^{2}}^{2}=2$

which contradicts (6.5). Now, choose $g\in G$ such that
$\prod_{i=1}^{\infty}\omega^{g_{i}}\in U$ (such $g$ must exist by density).
Then if we take $s=\omega^{g}$, equation (6.4) implies that
$\sigma^{p}(g,\cdot)$ is a constant. As $\sigma^{p}(g,t)$ clearly depends on
$t$, this is a contradiction. ∎

We now complete the proof of Theorem 6.2. Let $B\subseteq C_{p}$ be as in
Lemma 6.4. Let $\pi:C_{p^{2}}\rightarrow C_{p}$ be the map
$\pi_{i}(x)=x_{1}^{p}$ and let $A=\mathcal{T}\times\widetilde{B}$ where
$\widetilde{B}=\pi^{-1}(B)$. Then $\mu_{X}(A)=\frac{|B|}{p}$, and as in the
proof of Lemma 6.5,

$\mu_{X}(A\cap T_{apg}A\cap
T_{bpg}A)=\frac{|B|}{p^{2}}=\frac{p^{l-2}}{|B|^{l-1}}\mu_{X}(A)^{l}<\mu_{X}(A)^{l}.$

This completes the proof. ∎

##  7\. 3-point configurations in $\mathbb{Z}^{2}$

In this section, we establish ergodic popular difference densities for all
3-point matrix patterns in $\mathbb{Z}^{2}$. The results are summarized in
Table 1.1 in the introduction.

###  7.1. Ergodic popular difference densities when $r(M_{1},M_{2})=(2,1,1)$

The following Theorem gives an affirmative answer to Question 1.12 for the
group $G=\mathbb{Z}^{2}$:

######  Theorem 7.1.

Suppose $M_{1}$ and $M_{2}$ are $2\times 2$ matrices such that
$r(M_{1},M_{2})=(2,1,1)$. Then for any $\alpha\in(0,1)$,
$\text{epdd}_{M_{1},M_{2}}(\alpha)=\alpha^{3}$.

An example of the configurations handled by Theorem 7.1 is the class of all
axis-aligned right triangles in $\mathbb{Z}^{2}$,
$\\{(a,b),(a+n,b),(a,b+m)\\}$, which corresponds to the choice of matrices

$\displaystyle M_{1}=\left(\begin{array}[]{cc}1&0\\\
0&0\end{array}\right)\qquad\text{and}\qquad
M_{2}=\left(\begin{array}[]{cc}0&0\\\ 0&1\end{array}\right).$

###### Proof of Theorem 7.1.

Without loss of generality, we may assume
$\text{rk}(M_{1})=\text{rk}(M_{2})=1$ and $\text{rk}(M_{2}-M_{1})=2$. Indeed,
if $\text{rk}(M_{1})=2$, we may rearrange the expression

$\displaystyle\mu\left(A\cap T_{M_{1}\vec{n}}^{-1}A\cap
T_{M_{2}\vec{n}}^{-1}A\right)=\mu\left(A\cap
T_{(M_{1}-M_{2})\vec{n}}^{-1}A\cap T_{-M_{2}\vec{n}}^{-1}A\right)$

and the new matrices $N_{1}=M_{1}-M_{2}$ and $N_{2}=-M_{2}$ satisfy the
desired conditions.

We now break the proof into two cases depending on the diagonalizability of
$M_{1}$ and $M_{2}$. Note that, since $M_{i}$ has rank 1, its characteristic
polynomial is of the form $x(x-a)$ for some $a\in\mathbb{Z}$. Hence, if
$M_{i}$ has a nonzero eigenvalue, then it has an integer eigenvalue (in this
case, equal to $a$) and is diagonalizable.

Case 1: $M_{1}$ or $M_{2}$ has a nonzero eigenvalue.

Without loss of generality, we may assume that $M_{1}$ has a nonzero
eigenvalue and is therefore diagonalizable. Hence, there is a nonsingular
$2\times 2$ integer matrix $P$, an integer $a\in\mathbb{Z}$, and a rank 1
matrix $N_{2}$ with integer entries such that

$\displaystyle M_{1}P=P\left(\begin{array}[]{cc}a&0\\\
0&0\end{array}\right),\qquad
M_{2}P=PN_{2},\qquad\text{and}\qquad\text{rk}\left(N_{2}-\left(\begin{array}[]{cc}a&0\\\
0&0\end{array}\right)\right)=2.$

It is straightforward to check that, in order to satisfy the constraints on
rank, $N_{2}$ must be of the form

$\displaystyle N_{2}=\left(\begin{array}[]{cc}cd&c\\\ bd&b\end{array}\right)$

with $b\neq 0$. By changing to the basis $\binom{1}{-d},\binom{0}{1}$, we may
further assume $d=0$.

Suppose
$\left(X,\mathcal{X},\mu,(T_{\vec{n}})_{\vec{n}\in\mathbb{Z}^{2}}\right)$ is a
measure-preserving $\mathbb{Z}^{2}$-system (we do not need to assume that the
system is ergodic here), and let $A\in\mathcal{X}$ with $\mu(A)=\alpha$.
Define a new $\mathbb{Z}^{2}$-action by $S_{\vec{n}}:=T_{P\vec{n}}$. Then

$\displaystyle\text{UC~{}-}\lim_{\vec{n}\in\mathbb{Z}^{2}}{\mu\left(A\cap
T_{M_{1}P\vec{n}}^{-1}A\cap
T_{M_{2}P\vec{n}}^{-1}A\right)}=\text{UC~{}-}\lim_{\vec{n}\in\mathbb{Z}^{2}}{\mu\left(A\cap
S_{(an_{1},0)}^{-1}A\cap S_{(cn_{2},bn_{2})}^{-1}A\right)}.$

Now put $S_{1}:=S_{(a,0)}$ and $S_{2}:=S_{(c,b)}$. By Lemma 2.2 and the mean
ergodic theorem, we have

$\displaystyle\text{UC~{}-}\lim_{\vec{n}\in\mathbb{Z}^{2}}{\mu\left(A\cap
T_{M_{1}P\vec{n}}^{-1}A\cap T_{M_{2}P\vec{n}}^{-1}A\right)}$
$\displaystyle=\text{UC~{}-}\lim_{n_{2}\in\mathbb{Z}}{\text{UC~{}-}\lim_{n_{1}\in\mathbb{Z}}{\mu\left(A\cap
S_{1}^{-n_{1}}A\cap S_{2}^{-n_{2}}A\right)}}$
$\displaystyle=\int_{X}{\mathbbm{1}_{A}\cdot\operatorname{\mathbb{E}}(\mathbbm{1}_{A}\mid\mathcal{I}(S_{1}))\cdot\operatorname{\mathbb{E}}(\mathbbm{1}_{A}\mid\mathcal{I}(S_{2}))}$
$\displaystyle\geq\alpha^{3}.$

Therefore, for any $\varepsilon>0$, the set

$\displaystyle
R_{\varepsilon}:=\left\\{\vec{n}\in\mathbb{Z}^{2}:\mu\left(A\cap
T_{M_{1}P\vec{n}}^{-1}A\cap
T_{M_{2}P\vec{n}}^{-1}A\right)>\alpha^{3}-\varepsilon\right\\}$

is syndetic. Noting that $P$ is nonsingular, it follows that the set
$P(R_{\varepsilon})$ is also syndetic in $\mathbb{Z}^{2}$. But for any
$\vec{m}\in P(R_{\varepsilon})$, we have

$\displaystyle\mu\left(A\cap T_{M_{1}\vec{m}}^{-1}A\cap
T_{M_{2}\vec{m}}^{-1}A\right)>\alpha^{3}-\varepsilon.$

This shows $\text{epdd}_{M_{1},M_{2}}(\alpha)\geq\alpha^{3}$.

To see the upper bound $\text{epdd}_{M_{1},M_{2}}(\alpha)\leq\alpha^{3}$, let
$\left(X,\mathcal{X},\mu,(T_{\vec{n}})_{\vec{n}\in\mathbb{Z}^{2}}\right)$ be
mixing of order 3. Then for any $A\in\mathcal{X}$, we have $\mu\left(A\cap
T_{\vec{n}}^{-1}A\cap T_{\vec{m}}^{-1}A\right)\to\mu(A)^{3}$ as
$\vec{n},\vec{m},\vec{m}-\vec{n}\to\infty$. Let $P$ be a nonsingular $2\times
2$ matrix with integer entries and $a,b,c\in\mathbb{Z}$ with $a,b\neq 0$ such
that

$\displaystyle PM_{1}=\left(\begin{array}[]{cc}a&0\\\
0&0\end{array}\right)P,\qquad\text{and}\qquad
PM_{2}=\left(\begin{array}[]{cc}0&c\\\ 0&b\end{array}\right)P.$

The group of transformations $\widetilde{T}_{\vec{n}}:=T_{P\vec{n}}$ is still
mixing of order 3. Write $\vec{m}=P\vec{n}$ for $\vec{n}\in\mathbb{Z}^{2}$. If
$m_{1}\to\infty$ and $m_{2}\to\infty$, then

$\displaystyle\mu\left(A\cap\widetilde{T}_{M_{1}\vec{n}}^{-1}A\cap\widetilde{T}_{M_{2}\vec{n}}^{-1}A\right)=\mu\left(A\cap
T_{(am_{1},0)}^{-1}A\cap T_{(cm_{2},bm_{2})}^{-1}A\right)\to\mu(A)^{3}.$

Hence, for any $\varepsilon>0$, there is a finite set $F\subseteq\mathbb{Z}$
such that

$\displaystyle\left\\{\vec{n}\in\mathbb{Z}^{2}:\mu\left(A\cap\widetilde{T}_{M_{1}\vec{n}}^{-1}A\cap\widetilde{T}_{M_{2}\vec{n}}^{-1}A\right)>\mu(A)^{3}+\varepsilon\right\\}\subseteq\left\\{\vec{n}\in\mathbb{Z}^{2}:P\vec{n}\in(F\times\mathbb{Z})\cup(\mathbb{Z}\times
F)\right\\}.$

A union of finitely many lines in $\mathbb{Z}^{2}$ is not syndetic, so

$\displaystyle\text{synd-
sup}_{\vec{n}\in\mathbb{Z}^{2}}{\mu\left(A\cap\widetilde{T}_{M_{1}\vec{n}}^{-1}A\cap\widetilde{T}_{M_{2}\vec{n}}^{-1}A\right)}\leq\mu(A)^{3}.$

Case 2: $M_{1}$ and $M_{2}$ have no nonzero eigenvalues.

Since $M_{1}$ has rank 1, there is a nonsingular $2\times 2$ integer matrix
$P$, a nonzero integer $a\in\mathbb{Z}$, and a rank 1 matrix $N_{2}$ with
integer entries and characteristic polynomial $x^{2}$ such that

$\displaystyle M_{1}P=P\left(\begin{array}[]{cc}0&a\\\
0&0\end{array}\right),\qquad
M_{2}P=PN_{2},\qquad\text{and}\qquad\text{rk}\left(N_{2}-\left(\begin{array}[]{cc}0&a\\\
0&0\end{array}\right)\right)=2.$

Write

$\displaystyle N_{2}=\left(\begin{array}[]{cc}s&t\\\ u&v\end{array}\right).$

Since $N_{2}$ has characteristic polynomial $x^{2}$, we have $s+v=0$ and
$sv=tu$. Therefore, if $u=0$, then $s=v=0$. But then

$\displaystyle N_{2}-\left(\begin{array}[]{cc}0&a\\\
0&0\end{array}\right)=\left(\begin{array}[]{cc}0&t-a\\\ 0&0\end{array}\right)$

has rank at most 1. Thus, we must have $u\neq 0$. It follows that $N_{2}$ can
be written in the form

$\displaystyle N_{2}=\left(\begin{array}[]{cc}db&-d^{2}b\\\
b&-db\end{array}\right)$

for some $b,d$ with $b\neq 0$. Changing to the basis
$\binom{1}{0},\binom{d}{1}$, we may assume $d=0$ so that

$\displaystyle N_{2}=\left(\begin{array}[]{cc}0&0\\\ b&0\end{array}\right).$

Given a $\mathbb{Z}^{2}$-system
$\left(X,\mathcal{X},\mu,(T_{\vec{n}})_{\vec{n}\in\mathbb{Z}^{2}}\right)$,
note that

$\displaystyle\mu\left(A\cap T_{N_{1}\vec{n}}^{-1}A\cap
T_{N_{2}\vec{n}}^{-1}A\right)=\mu\left(A\cap T_{(an_{2},0)}^{-1}A\cap
T_{(0,bn_{1})}^{-1}A\right).$

Hence, replacing $(n_{1},n_{2})$ by $(n_{2},n_{1})$, we reduce to Case 1. ∎

###  7.2. Ergodic popular difference densities when $r(M_{1},M_{2})=(1,1,1)$

For matrix configurations with $r(M_{1},M_{2})=(1,1,1)$, we must distinguish
between several cases. First, when $M_{1}$ and $M_{2}$ commute, a construction
based on Behrend’s theorem shows that the ergodic popular difference density
decays faster than any polynomial:

######  Theorem 7.2.

Suppose $M_{1}$ and $M_{2}$ are commuting $2\times 2$ matrices such that
$r(M_{1},M_{2})=(1,1,1)$. Then for any sufficiently small $\alpha\in(0,1)$,
$\text{epdd}_{M_{1},M_{2}}(\alpha)<\alpha^{c\log(1/\alpha)}$, where $c>0$ is
an absolute constant.

Theorem 7.2 applies to collinear three-point configurations up to scaling and
translation.

###### Proof of Theorem 7.2.

We first distinguish between two cases depending on diagonalizability of
$M_{1}$ and $M_{2}$.

Case 1: $M_{1}$ or $M_{2}$ has a nonzero eigenvalue.

Without loss of generality, assume $M_{1}$ has a nonzero eigenvalue and is
therefore diagonalizable. Since $M_{2}$ and $M_{2}-M_{1}$ are also rank 1 and
commute with $M_{1}$, there exists a nonsingular $2\times 2$ matrix $P$ with
integer entries and $a,b\in\mathbb{Z}$ be distinct and nonzero such that

(7.5) $\displaystyle PM_{1}=\left(\begin{array}[]{cc}a&0\\\
0&0\end{array}\right)P\qquad\text{and}\qquad
PM_{2}=\left(\begin{array}[]{cc}b&0\\\ 0&0\end{array}\right)P.$

Case 2: $M_{1}$ and $M_{2}$ have no nonzero eigenvalues.

Using the condition $r(M_{1},M_{2})=(1,1,1)$, there is a nonsingular $2\times
2$ integer matrix $P$, a nonzero integer $a\in\mathbb{Z}$, and a rank 1 matrix
$N_{2}$ with integer entries and characteristic polynomial $x^{2}$ such that

$\displaystyle M_{1}P=P\left(\begin{array}[]{cc}0&a\\\
0&0\end{array}\right),\qquad
M_{2}P=PN_{2},\qquad\text{and}\qquad\text{rk}\left(N_{2}-\left(\begin{array}[]{cc}0&a\\\
0&0\end{array}\right)\right)=1.$

Moreover, $N_{2}$ commutes with the matrix $\left(\begin{array}[]{cc}0&a\\\
0&0\end{array}\right)$. Write

$\displaystyle N_{2}=\left(\begin{array}[]{cc}s&t\\\ u&v\end{array}\right).$

Note that

$\displaystyle\left[\left(\begin{array}[]{cc}0&a\\\
0&0\end{array}\right),\left(\begin{array}[]{cc}s&t\\\
u&v\end{array}\right)\right]=\left(\begin{array}[]{cc}au&a(v-s)\\\
0&au\end{array}\right),$

so $u=0$ and $v=s$. On the other hand, since $N_{2}$ has characteristic
polynomial $x^{2}$, we have $s+v=0$ and $sv=tu$. Hence, $s=v=0$, and $N_{2}$
is of the form

$\displaystyle N_{2}=\left(\begin{array}[]{cc}0&b\\\ 0&0\end{array}\right)$

with $b\notin\\{0,a\\}$.

Now, replacing $(n_{1},n_{2})\in\mathbb{Z}^{2}$ by
$(n_{2},n_{1})\in\mathbb{Z}^{2}$ and using the identity

$\displaystyle\left(\begin{array}[]{cc}0&c\\\
0&0\end{array}\right)\left(\begin{array}[]{c}n_{2}\\\
n_{1}\end{array}\right)=\left(\begin{array}[]{cc}c&0\\\
0&0\end{array}\right)\left(\begin{array}[]{c}n_{1}\\\ n_{2}\end{array}\right)$

for $c\in\mathbb{Z}$, we can reduce Case 2 to Case 1.

Without loss of generality, let $P$ be a nonsingular $2\times 2$ matrix with
integer entries and $a,b\in\mathbb{Z}$ distinct and nonzero such that (7.5)
holds. Put $d:=\left|\det(P)\right|\in\mathbb{N}$.

Define $S:\mathbb{T}^{2}\to\mathbb{T}^{2}$ by $S(x,y):=(x,y+x)$. Let
$R:\mathbb{T}^{2}\to\mathbb{T}^{2}$ be the transformation $R(x,y)=(2x,2y+x)$.
Both $S$ and $R$ preserve the Haar probability measure $\mu$ on
$\mathbb{T}^{2}$. We claim that the $(\mathbb{Z}_{\geq 0})^{2}$-action
generated by $S$ and $R$ is ergodic (with respect to $\mu$). To see this,
suppose $f\in L^{2}(\mathbb{T}^{2})$ is simultaneously $S$\- and
$R$-invariant, and expand $f$ as a Fourier series

$\displaystyle f(x,y)=\sum_{n,m}{c_{n,m}e(nx+my)},$

where $e(t):=e(2\pi it)$. Then

$\displaystyle(Sf)(x,y)=\sum_{n,m}{c_{n,m}e((n+m)x+my)}=\sum_{n,m}{c_{n-m,m}e(nx+my)}.$

Therefore, since $Sf=f$, we have $c_{n,m}=c_{n-m,m}$ for all
$n,m\in\mathbb{Z}$. By Parseval’s identity,
$\sum_{n,m}{|c_{n,m}|^{2}}=\|f\|_{2}^{2}<\infty$, so $c_{n,m}=0$ whenever
$m\neq 0$. That is, $f(x,y)=\sum_{n}{c_{n,0}e(nx)}$. Now,

$\displaystyle(Rf)(x,y)=\sum_{n}{c_{n,0}e(2nx)}.$

Hence, since $Rf=f$, we have $c_{2n,0}=c_{n,0}$ for every $n\in\mathbb{Z}$.
Applying Parseval’s identity once again, we conclude that $c_{n,0}=0$ for
$n\neq 0$. Thus, $f(x,y)=c_{0,0}$ is a constant function.

Fix $\alpha\in(0,1)$. By [BHK05, Theorem 1.3], there exists a set
$A\subseteq\mathbb{T}^{2}$ with $\mu(A)=\alpha$ such that $\mu\left(A\cap
S^{-an}A\cap S^{-bn}A\right)<\alpha^{c\log(1/\alpha)}$ for $n\neq 0$, where
$c>0$ is an absolute constant.555The statement of [BHK05, Theorem 1.3] only
gives a bound of the form $\alpha^{l}$ rather than $\alpha^{c\log(1/\alpha)}$.
However, as noted in [BHK05] immediately after the statement, the construction
of the set $A$ gives this stronger bound via Behrend’s theorem on sets without
three-term arithmetic progressions [Beh46]. Additionally, [BHK05, Theorem 1.3]
is only stated for the case $a=1,b=2$, but the same method works for general
$a,b$; see, e.g., [ABB21, Section 11].

Let $\left(X,\mathcal{X},\nu,(T_{\vec{n}})_{\vec{n}\in\mathbb{Z}^{2}}\right)$
be an ergodic $\mathbb{Z}^{2}$-system and $B\in\mathcal{X}$ with
$\nu(B)=\alpha$ such that

$\displaystyle\nu\left(B\cap T_{\vec{n}}^{-1}B\cap
T_{\vec{m}}^{-1}B\right)=\mu\left(A\cap S^{-n_{1}}R^{-n_{2}}A\cap
S^{-m_{1}}R^{-m_{2}}A\right)$

for every $\vec{n},\vec{m}\in\mathbb{Z}\times\mathbb{Z}_{\geq 0}$. (Note that,
because $R$ is non-invertible, we cannot simply take $X=\mathbb{T}^{2}$,
$\nu=\mu$, $B=A$, and $T_{\vec{n}}=S^{n_{1}}R^{n_{2}}$.) Then let
$\widetilde{T}_{\vec{n}}:=T_{P\vec{n}}$ for $\vec{n}\in\mathbb{Z}^{2}$.

Since $[\mathbb{Z}^{2}:P(\mathbb{Z}^{2})]=\left|\det(P)\right|=d<\infty$, the
system
$\left(X,\mathcal{X},\nu,(\widetilde{T}_{\vec{n}})_{\vec{n}\in\mathbb{Z}^{2}}\right)$
has at most $d$ ergodic components. Hence, we may write the ergodic
decomposition as $\nu=\frac{1}{k}\sum_{i=1}^{k}{\nu_{i}}$ for some $k\leq d$
and some measure $\nu_{i}$. For some $1\leq i\leq k$, we must have
$\nu_{i}(B)\geq\alpha$. Without loss of generality, we may therefore assume
$\nu_{1}(B)\geq\alpha$.

Let $\vec{n}\in\mathbb{Z}^{2}\setminus\\{0\\}$. Let
$\vec{m}=P\vec{n}\in\mathbb{Z}^{2}$. Then

$\displaystyle\nu_{1}\left(B\cap\widetilde{T}_{M_{1}\vec{n}}^{-1}B\cap\widetilde{T}_{M_{2}\vec{n}}^{-1}B\right)$
$\displaystyle=\nu_{1}\left(B\cap T_{(am_{1},0)}^{-1}B\cap
T_{(bm_{1},0)}^{-1}B\right)$ $\displaystyle\leq d\cdot\mu\left(A\cap
S^{-am_{1}}A\cap S^{-bm_{1}}A\right).$

Hence, if
$\nu_{1}\left(B\cap\widetilde{T}_{M_{1}\vec{n}}^{-1}B\cap\widetilde{T}_{M_{2}\vec{n}}^{-1}B\right)>d\cdot\alpha^{c\log(1/\alpha)}$,
then $m_{1}=0$. But since $P$ is nonsingular,

$\displaystyle\left\\{\vec{n}\in\mathbb{Z}^{2}:P\vec{n}\in\\{0\\}\times\mathbb{Z}\right\\}=\mathbb{Q}\vec{v}\cap\mathbb{Z}^{2}$

where $\vec{v}$ is the vector $P^{-1}\binom{0}{1}\in\mathbb{Q}^{2}$. Such a
set is never syndetic, so $\text{epdd}_{M_{1},M_{2}}(\alpha)\leq
d\cdot\alpha^{c\log(1/\alpha)}$. For $c^{\prime}<c$ and $\alpha$ sufficiently
small, one has
$d\cdot\alpha^{c\log(1/\alpha)}<\alpha^{c^{\prime}\log(1/\alpha)}$, so this
completes the proof. ∎

Now suppose $r(M_{1},M_{2})=(1,1,1)$, and $M_{1}$ and $M_{2}$ do not commute.
In this case, $M_{1}$ or $M_{2}$ must be diagonalizable,666If neither $M_{1}$
nor $M_{2}$ are diagonalizable, then they both have characteristic polynomial
$x^{2}$. By a change of basis, we may assume $M_{1}$ is in its Jordan form
$M_{1}=\left(\begin{array}[]{cc}0&1\\\ 0&0\end{array}\right)$. Write
$M_{2}=\left(\begin{array}[]{cc}a&b\\\ c&d\end{array}\right)$. The condition
$\text{rk}(M_{2})=\text{rk}(M_{2}-M_{1})=1$ implies that $ad-bc=ad-(b-1)c=0$,
so $c=0$ and $ad=0$. Moreover, since $M_{2}$ has characteristic polynomial
$x^{2}$, we have $a+d=0$. Hence, $M_{2}=\left(\begin{array}[]{cc}0&b\\\
0&0\end{array}\right)$. But then $M_{2}$ commutes with $M_{1}$. so we assume
without loss of generality that $M_{1}$ is diagonalizable. We then distinguish
between two cases, depending on the form of $M_{2}$ when $M_{1}$ is
diagonalized. Call the pair of matrices $(M_{1},M_{2})$ _row-like_ if there is
a non-singular $2\times 2$ matrix $P$ with rational entries and rational
numbers $a,b,c\in\mathbb{Q}$ with $a,b\neq 0$ such that

$\displaystyle PM_{1}P^{-1}=\left(\begin{array}[]{cc}a&0\\\
0&0\end{array}\right)\qquad\text{and}\qquad
PM_{2}P^{-1}=\left(\begin{array}[]{cc}c&b\\\ 0&0\end{array}\right).$

Similarly, call the pair $(M_{1},M_{2})$ _column-like_ if there is a non-
singular $2\times 2$ matrix $P$ with rational entries and rational numbers
$a,b,c\in\mathbb{Q}$ with $a,b\neq 0$ such that

$\displaystyle PM_{1}P^{-1}=\left(\begin{array}[]{cc}a&0\\\
0&0\end{array}\right)\qquad\text{and}\qquad
PM_{2}P^{-1}=\left(\begin{array}[]{cc}c&0\\\ b&0\end{array}\right).$

For row-like configurations, we can use the “Fubini” property of uniform
Cesàro limits (Lemma 2.2) to show $\text{epdd}(\alpha)=\alpha^{3}$:

######  Theorem 7.3.

Suppose $M_{1}$ and $M_{2}$ are $2\times 2$ matrices with
$r(M_{1},M_{2})=(1,1,1)$ such that $(M_{1},M_{2})$ is row-like. Then for any
$\alpha\in(0,1)$, $\text{epdd}_{M_{1},M_{2}}(\alpha)=\alpha^{3}$.

###### Proof.

Let $P$ be a nonsingular $2\times 2$ matrix with integer entries such that

$\displaystyle M_{1}P=P\left(\begin{array}[]{cc}a&0\\\
0&0\end{array}\right)\qquad\text{and}\qquad
M_{2}P=P\left(\begin{array}[]{cc}c&b\\\ 0&0\end{array}\right).$

By changing to the basis $\binom{b}{-c},\binom{0}{1}$, we may assume $c=0$.

Let $\left(X,\mathcal{X},\mu,(T_{\vec{n}})_{\vec{n}\in\mathbb{Z}^{2}}\right)$
be a measure-preserving system, and let $A\in\mathcal{X}$ with
$\mu(A)=\alpha>0$. Define a new $\mathbb{Z}^{2}$-action by
$\widetilde{T}_{\vec{n}}:=T_{P\vec{n}}$, and let $S:=\widetilde{T}_{(1,0)}$.
Then

$\displaystyle\mu\left(A\cap T_{M_{1}P\vec{n}}^{-1}A\cap
T_{M_{2}P\vec{n}}^{-1}A\right)=\mu\left(A\cap S^{-an_{1}}A\cap
S^{-bn_{2}}A\right).$

Thus, by Lemma 2.2, we have

$\displaystyle\text{UC~{}-}\lim_{\vec{n}\in\mathbb{Z}^{2}}{\mu\left(A\cap
T_{M_{1}P\vec{n}}^{-1}A\cap T_{M_{2}P\vec{n}}^{-1}A\right)}\geq\alpha^{3}.$

Since $P$ is nonsingular, it follows that

$\displaystyle\text{synd-sup}_{\vec{n}\in\mathbb{Z}^{2}}{\mu\left(A\cap
T_{M_{1}\vec{n}}^{-1}A\cap T_{M_{2}\vec{n}}^{-1}A\right)}\geq\alpha^{3}.$

Now we will show $\text{epdd}_{M_{1},M_{2}}(\alpha)\leq\alpha^{3}$. Let $P$ be
a nonsingular $2\times 2$ matrix with integer entries and $a,b,c\in\mathbb{Z}$
with $a,b\neq 0$ such that

$\displaystyle PM_{1}=\left(\begin{array}[]{cc}a&0\\\
0&0\end{array}\right)P,\qquad\text{and}\qquad
PM_{2}=\left(\begin{array}[]{cc}0&b\\\ 0&0\end{array}\right)P.$

Let $\left(X,\mathcal{X},\mu,S,R\right)$ be an ergodic $\mathbb{Z}^{2}$-system
such that $S$ is mixing of order 3. Define $T_{\vec{n}}:=S^{n_{1}}R^{n_{2}}$
and $\widetilde{T}_{\vec{n}}:=T_{P\vec{n}}$ for $\vec{n}\in\mathbb{Z}^{2}$.
Then for $A\in\mathcal{X}$ and $\vec{m}=P\vec{n}\in\mathbb{Z}^{2}$, we have

$\displaystyle\mu\left(A\cap\widetilde{T}_{M_{1}\vec{n}}^{-1}A\cap\widetilde{T}_{M_{2}\vec{n}}^{-1}A\right)=\mu\left(A\cap
S^{-am_{1}}A\cap S^{-bm_{2}}A\right).$

Since $S$ is mixing of order 3, given $\varepsilon>0$, there exists a finite
set $F\subseteq\mathbb{Z}$ such that

$\displaystyle\left\\{\vec{n}\in\mathbb{Z}^{2}:\mu\left(A\cap\widetilde{T}_{M_{1}\vec{n}}^{-1}A\cap\widetilde{T}_{M_{2}\vec{n}}^{-1}A\right)>\mu(A)^{3}+\varepsilon\right\\}$
$\displaystyle\quad\subseteq
P^{-1}\left(\left\\{\vec{m}\in\mathbb{Z}^{2}:m_{1}\in F,m_{2}\in
F,~{}\text{or}~{}bm_{2}-am_{1}\in F\right\\}\right).$

This set is a union of finitely many lines in $\mathbb{Z}^{2}$, so it is not
syndetic. Hence,

$\displaystyle\text{synd-
sup}_{\vec{n}\in\mathbb{Z}^{2}}{\mu\left(A\cap\widetilde{T}_{M_{1}\vec{n}}^{-1}A\cap\widetilde{T}_{M_{2}\vec{n}}^{-1}A\right)}\leq\mu(A)^{3}.$

∎

The prototypical column-like configuration is the class of axis-aligned
isosceles right triangles, for which it is known by previous work of Chu [C11]
and Donoso and Sun [DS18] that
$\alpha^{4}\leq\text{epdd}(\alpha)\leq\alpha^{4-o(1)}$. We prove that these
bounds extend to all column-like configurations:

######  Theorem 7.4.

Suppose $M_{1}$ and $M_{2}$ are $2\times 2$ matrices with
$r(M_{1},M_{2})=(1,1,1)$ such that $(M_{1},M_{2})$ is column-like. Then for
any $\alpha\in(0,1)$, $\text{epdd}_{M_{1},M_{2}}(\alpha)\geq\alpha^{4}$.
Moreover, for any $l<4$ and all sufficiently small $\alpha$ (depending on
$l$), one has $\text{epdd}_{M_{1},M_{2}}(\alpha)\leq\alpha^{l}$.

###### Proof.

Let $\left(X,\mathcal{X},\mu,(T_{\vec{n}})_{\vec{n}\in\mathbb{Z}^{2}}\right)$
be an ergodic $\mathbb{Z}^{2}$-system. Since the pair $(M_{1},M_{2})$ is
column-like, there exists a nonsingular $2\times 2$ matrix $P$ with integer
entries and integers $a,b,c\in\mathbb{Z}$ with $a,b\neq 0$ such that

$\displaystyle M_{1}P=P\left(\begin{array}[]{cc}a&0\\\
0&0\end{array}\right)\qquad\text{and}\qquad
M_{2}P=P\left(\begin{array}[]{cc}c&0\\\ b&0\end{array}\right).$

Then for any $\vec{n}\in\mathbb{Z}^{2}$, we have

$\displaystyle\mu\left(A\cap T_{M_{1}P\vec{n}}^{-1}A\cap
T_{M_{2}P\vec{n}}^{-1}A\right)=\mu\left(A\cap T_{P(an_{1},0)}^{-1}A\cap
T_{P(cn_{1},bn_{1})}^{-1}A\right).$

Letting $S:=T_{P(a,0)}$ and $R:=T_{P(c,b)}$, we therefore have the identity

$\displaystyle\mu\left(A\cap T_{M_{1}P\vec{n}}^{-1}A\cap
T_{M_{2}P\vec{n}}^{-1}A\right)=\mu\left(A\cap S^{-n_{1}}A\cap
R^{-n_{1}}A\right).$

Now, since $T$ is ergodic and $P$ is nonsingular, the $\mathbb{Z}^{2}$-action
generated by $S$ and $R$ has finitely many ergodic components. Thus, by [C11,
Theorem 1.1],

$\displaystyle\left\\{n\in\mathbb{Z}:\mu\left(A\cap S^{-n}A\cap
R^{-n}A\right)\geq\mu(A)^{4}\right\\}$

is syndetic in $\mathbb{Z}$.777In [C11], it is assumed that the system
$(X,\mathcal{X},\mu,S,R)$ is ergodic. However, the proof easily extends to the
case that the system has finitely many ergodic components by noting that all
of the ergodic components will have the same Kronecker factor. It follows that

$\displaystyle\left\\{\vec{n}\in\mathbb{Z}^{2}:\mu\left(A\cap
T_{M_{1}\vec{n}}^{-1}A\cap
T_{M_{2}\vec{n}}^{-1}A\right)\geq\mu(A)^{4}\right\\}$

is syndetic in $\mathbb{Z}^{2}$. Hence,
$\text{epdd}_{M_{1},M_{2}}(\alpha)\geq\alpha^{4}$.

Let $l<4$. By [DS18, Theorem 1.2], there exists an ergodic
$\mathbb{Z}^{2}$-system $\left(X,\mathcal{X},\mu,S,R\right)$ and a set
$A\in\mathcal{X}$ such that $\mu\left(A\cap S^{-n}A\cap
R^{-n}A\right)<\mu(A)^{l}$ for every $n\neq 0$. Since the pair $(M_{1},M_{2})$
is column-like, there is a nonsingular $2\times 2$ matrix $P$ with integer
entries and integers $a,b,c\in\mathbb{Z}$ with $a,b\neq 0$ such that

$\displaystyle PM_{1}=\left(\begin{array}[]{cc}a&0\\\
0&0\end{array}\right)P\qquad\text{and}\qquad
PM_{2}=\left(\begin{array}[]{cc}c&0\\\ b&0\end{array}\right)P.$

Define $T_{\vec{n}}:=S^{bn_{1}}(R^{a}S^{-c})^{n_{2}}$, and let
$\widetilde{T}_{\vec{n}}:=T_{P\vec{n}}$ for $n\in\mathbb{Z}^{2}$. Note that
$\left(X,\mathcal{X},\mu,(\widetilde{T}_{\vec{n}})_{\vec{n}\in\mathbb{Z}^{2}}\right)$
has finitely many ergodic components. To be more precise, the ergodic
decomposition has the form $\mu=\frac{1}{k}\sum_{i=1}^{k}{\mu_{i}}$ with
$k\leq d:=\left|ab\det(P)\right|$. Without loss of generality, we may assume
$\mu_{1}(A)\geq\mu(A)$.

Now, for any $\vec{n}\neq 0$, we have

$\displaystyle\mu_{1}\left(A\cap\widetilde{T}_{M_{1}\vec{n}}^{-1}A\cap\widetilde{T}_{M_{2}\vec{n}}^{-1}A\right)\leq
d\cdot\mu\left(A\cap S^{-abm_{1}}A\cap R^{-abm_{1}}A\right)$

where $\vec{m}=P\vec{n}\in\mathbb{Z}^{2}$. Therefore,

$\displaystyle\left\\{\vec{n}\in\mathbb{Z}^{2}:\mu_{1}\left(A\cap\widetilde{T}_{M_{1}\vec{n}}^{-1}A\cap\widetilde{T}_{M_{2}\vec{n}}^{-1}A\right)\geq
d\cdot\mu_{1}(A)^{l}\right\\}\subseteq\left\\{\vec{n}\in\mathbb{Z}^{2}:P\vec{n}\in\\{0\\}\times\mathbb{Z}\right\\}\subseteq\mathbb{Q}\vec{v}\cap\mathbb{Z}^{2},$

where $\vec{v}=P^{-1}\binom{0}{1}\in\mathbb{Q}^{2}$. The set
$\mathbb{Q}\vec{v}\cap\mathbb{Z}^{2}$ is not syndetic, so this shows
$\text{epdd}_{M_{1},M_{2}}(\alpha)\leq d\cdot\alpha^{l}$ for $\alpha=\mu(A)$.
Moreover, for any $l^{\prime}<l$, we have the inequality
$d\cdot\alpha^{l}<\alpha^{l^{\prime}}$ for all $\alpha>0$ sufficiently small.
∎

###  7.3. Finitary combinatorial consequences and open questions

There are two cases in which our ergodic-theoretic results directly imply
finitary combinatorial analogues. Namely, when $r(M_{1},M_{2})=(2,1,1)$ and
when $(M_{1},M_{2})$ is a row-like pair of non-commuting matrices with
$r(M_{1},M_{2})=(1,1,1)$, we establish the bound
$\text{epdd}_{M_{1},M_{2}}(\alpha)\geq\alpha^{3}$ with the help of the
“Fubini” property for uniform Cesàro limits (Lemma 2.2), and this allows us to
avoid assuming that the underlying $\mathbb{Z}^{2}$-system is ergodic. For
this reason, we can obtain the following combinatorial result:

######  Theorem 7.5.

Let $M_{1},M_{2}$ be $2\times 2$ matrices with integer entries. Suppose that
either

  1. (i)

$r(M_{1},M_{2})=(2,1,1)$, or

  2. (ii)

$r(M_{1},M_{2})=(1,1,1)$, $M_{1}$ and $M_{2}$ do not commute, and
$(M_{1},M_{2})$ is row-like.

Then for any $\alpha,\varepsilon>0$, there exists
$N_{0}=N_{0}(\alpha,\varepsilon)\in\mathbb{N}$ such that, if $N\geq N_{0}$ and
$A\subseteq\\{1,\dots,N\\}^{2}$ has $|A|\geq\alpha N^{2}$, then there exists
$\vec{n}\in\mathbb{Z}^{2}$ with
$M_{1}\vec{n},M_{2}\vec{n},(M_{2}-M_{1})\vec{n}\neq 0$ such that

$\displaystyle\left|\left\\{\vec{x}\in\mathbb{Z}^{2}:\\{\vec{x},\vec{x}+M_{1}\vec{n},\vec{x}+M_{2}\vec{n}\\}\subseteq
A\right\\}\right|>(\alpha^{3}-\varepsilon)N^{2}.$

###### Proof.

Let $\alpha,\varepsilon>0$ and suppose no such $N_{0}$ exists. Then there is
an increasing sequence $(N_{k})_{k\in\mathbb{N}}$ in $\mathbb{N}$ and sets
$A_{k}\subseteq\\{1,\dots,N_{k}\\}^{2}$ with $|A_{k}|\geq\alpha N_{k}^{2}$
such that

$\displaystyle\left|A_{k}\cap\left(A_{k}-M_{1}\vec{n}\right)\cap\left(A_{k}-M_{2}\vec{n}\right)\right|\leq(\alpha^{3}-\varepsilon)N_{k}^{2}$

whenever $M_{1}\vec{n},M_{2}\vec{n},(M_{2}-M_{1})\vec{n}\neq 0$.

For notational convenience, let $A_{k,0}:=\mathbb{Z}^{2}\setminus A_{k}$ and
$A_{k,1}:=A_{k}$. By passing to a subsequence if necessary, we may assume
without loss of generality that

(7.6)
$\displaystyle\lim_{k\to\infty}{\frac{\left|\left(A_{k,i_{1}}-\vec{n}_{1}\right)\cap\dots\cap\left(A_{k,i_{r}}-\vec{n}_{r}\right)\cap\\{1,\dots,N_{k}\\}^{2}\right|}{N_{k}^{2}}}$

exists for all $r\in\mathbb{N}$,
$\vec{n}_{1},\dots,\vec{n}_{r}\in\mathbb{Z}^{2}$, and
$i_{1},\dots,i_{r}\in\\{0,1\\}$. Hence, we may define a measure $\mu$ on the
sequence space $\\{0,1\\}^{\mathbb{Z}^{2}}$ by setting

$\displaystyle\mu\left(\left\\{x\in
X:x(\vec{n}_{1})=i_{1},\dots,x(\vec{n}_{r})=i_{r}\right\\}\right)$

equal to the limit (7.6) and extending with the use of Kolmogorov’s extension
theorem. Since $\left(\\{1,\dots,N_{k}\\}^{2}\right)_{k\in\mathbb{N}}$ is a
Følner sequence in $\mathbb{Z}^{2}$, the measure $\mu$ is invariant under the
shift transformations $(T_{\vec{n}}x)(\vec{m}):=x(\vec{m}+\vec{n})$.

Let $A:=\\{x\in X:x(\vec{0})=1\\}$. Then
$\mu(A)=\lim_{k\to\infty}{\frac{|A_{k}|}{N_{k}^{2}}}\geq\alpha$. On the other
hand, if $M_{1}\vec{n},M_{2}\vec{n},(M_{2}-M_{1})\vec{n}\neq 0$, then

$\displaystyle\mu\left(A\cap T_{M_{1}\vec{n}}^{-1}A\cap
T_{M_{2}\vec{n}}^{-1}A\right)$ $\displaystyle=\mu\left(\\{x\in
X:x(\vec{0})=x(M_{1}\vec{n})=x(M_{2}\vec{n})=1\\}\right)$
$\displaystyle=\lim_{k\to\infty}{\frac{\left|A_{k}\cap\left(A_{k}-M_{1}\vec{n}\right)\cap\left(A_{k}-M_{2}\vec{n}\right)\right|}{N_{k}^{2}}}$
$\displaystyle\leq\alpha^{3}-\varepsilon.$

Hence,

$\displaystyle R_{\varepsilon}$
$\displaystyle:=\left\\{\vec{n}\in\mathbb{Z}^{2}:\mu\left(A\cap
T_{M_{1}\vec{n}}^{-1}A\cap
T_{M_{2}\vec{n}}^{-1}A\right)>\mu(A)^{3}-\varepsilon\right\\}$
$\displaystyle\subseteq\ker(M_{1})\cup\ker(M_{2})\cup\ker(M_{2}-M_{1}).$

But by the proofs of Theorems 7.1 and 7.3, $R_{\varepsilon}$ is a syndetic
subset of $\mathbb{Z}^{2}$, so this is a contradiction. ∎

For general 3-point matrix patterns in $\mathbb{Z}^{2}$, it remains an open
problem to fully determine (finitary combinatorial) popular difference
densities. One particularly attractive case, which can be seen as a finitary
version of Question 1.12 for the group $G=\mathbb{Z}^{2}$, is the following:

######  Conjecture 7.6.

Let $M_{1}$ and $M_{2}$ be $2\times 2$ matrices with integer entries such that
$M_{2}-M_{1}$ has full rank. Then for any $\alpha,\varepsilon>0$, there exists
$N_{0}=N_{0}(\alpha,\varepsilon)\in\mathbb{N}$ such that, if $N\geq N_{0}$ and
$A\subseteq\\{1,\dots,N\\}^{2}$ has cardinality $|A|\geq\alpha N^{2}$, then
there exists $\vec{n}\in\mathbb{Z}^{2}$ with $M_{1}\vec{n},M_{2}\vec{n}\neq 0$
such that

$\displaystyle\left|\left\\{\vec{x}\in\mathbb{Z}^{2}:\\{\vec{x},\vec{x}+M_{1}\vec{n},\vec{x}+M_{2}\vec{n}\\}\subseteq
A\right\\}\right|>(\alpha^{3}-\varepsilon)N^{2}.$

The special case when $M_{1},M_{2}$, and $M_{2}-M_{1}$ are all invertible,
Conjecture 7.6 was verified by [BSST21, Theorem 1.1]. Moreover, Theorem 7.5
shows that Conjecture 7.6 holds when $M_{1}$ and $M_{2}$ are both rank 1
matrices. The most interesting remaining case is when $M_{1}$ has full rank
and $M_{2}$ is a rank 1 matrix.

Finally, the column-like family of configurations
$\\{(a,b),(a+n,b),(a,b+n)\\}$, known as _corners_ , has been well-studied from
the perspective of popular differences in finitary combinatorics. In
particular, it is known that the popular difference density for corners is of
the form $\alpha^{4-o(1)}$; see [Ber21] and also [M21, FSSSZ20] for an
analogous result in a finite characteristic setting. To the authors’
knowledge, such results are not known for general column-like matrix patterns,
but we anticipate that techniques for handling corners should apply in this
generality with only minor modifications needed.

##  8\. Khintchine-type recurrence for actions of semigroups

As a consequence of Theorem 1.13, we obtain the following combinatorial
result. For any set $E\subseteq\mathbb{Q}_{>0}$ of positive multiplicative
upper Banach density $d^{*}_{mult}(E)>0$ and any $\varepsilon>0$, there exists
$q\in\mathbb{Q}_{>0}\setminus\\{1\\}$ such that

$\displaystyle d^{*}_{mult}\left(E\cap q^{-1}E\cap
q^{-2}E\right)>d^{*}_{mult}(E)^{3}-\varepsilon$

(in fact, the set of such $q$ is multiplicatively syndetic). More generally,
for any countable field $\mathbb{F}$, any set $E\subseteq\mathbb{F}^{\times}$
of positive multiplicative upper Banach density $d_{mult}^{*}(E)>0$ and any
$\varepsilon>0$, the set of $x\in\mathbb{F}^{\times}$ such that

$\displaystyle d^{*}_{mult}\left(E\cap x^{-1}E\cap
x^{-2}E\right)>d^{*}_{mult}(E)^{3}-\varepsilon$

is multiplicatively syndetic.888In fact, our results show that for any
$k\in\mathbb{N}$, $d_{\text{mult}}^{*}\left(E\cap x^{-k}E\cap
x^{-(k+1)}E\right)$ and $d_{\text{mult}}^{*}\left(E\cap x^{-1}E\cap
x^{-k}E\right)$ can be made arbitrarily close to $d_{\text{mult}}^{*}(E)^{3}$
for a multiplicatively syndetic set of $x\in\mathbb{F}^{\times}$. On the other
hand, by Theorem 1.14, there are $n,m\in\mathbb{N}$ such that
$d_{\text{mult}}^{*}\left(E\cap x^{-n}E\cap x^{-m}E\right)$ is much smaller
than $d_{\text{mult}}^{*}(E)^{3}$ for all $x\neq 1$. This is suggestive of the
following problem. Let $R$ be an integral domain. (For example, $R$ can be the
ring $\mathbb{Z}$, the ring of integers of a number field, or the polynomial
ring $\mathbb{F}[t]$ over a finite field $\mathbb{F}$.) Given a set
$E\subseteq R^{\times}$ of positive multiplicative upper Banach density
$d^{*}_{R,mult}(E)>0$ and $\varepsilon>0$, does there exist $r\in
R\setminus\\{1\\}$ such that

$\displaystyle d^{*}_{R,mult}\left(E\cap E/r\cap
E/r^{2}\right)>d^{*}_{R,mult}(E)^{3}-\varepsilon,$

where $E/r:=\left\\{t\in R:rt\in E\right\\}$ for $r\in R$? The goal of this
section is to transfer our results into the setting of cancellative abelian
semigroups in order to answer this question affirmatively.

###  8.1. The group generated by a cancellative abelian semigroup

Let $(S,+)$ be a countable cancellative abelian semigroup. That is, $S$ is a
countable set equipped with a commutative and associative binary operation $+$
such that if $s+t=s+r$ for some $r,s,t\in S$, then $t=r$.

We can define a group $G_{S}$ as the set of formal differences
$\left\\{s-t:s,t\in S\right\\}$ where we identify $s-t$ and
$s^{\prime}-t^{\prime}$ if $s+t^{\prime}=s^{\prime}+t$. More formally, we may
define an equivalence relation $\sim$ on $S^{2}$ by
$(s,t)\sim(s^{\prime},t^{\prime})$ if $s+t^{\prime}=s^{\prime}+t$. Then
$G_{S}$ is the set of equivalence classes $S^{2}/\sim$ with the operation
$[(s,t)]+[(s^{\prime},t^{\prime})]:=[(s+s^{\prime},t+t^{\prime})]$. It is easy
to check that this operation is well-defined because $S$ is cancellative.
Moreover, $G_{S}$ has an identity $0:=[(s,s)]$, and for any $s,t\in S$, we
have $[(s,t)]+[(t,s)]=0$. Thus, $G_{S}$ is a group.

###  8.2. Notions of largeness

For a set $E\subseteq S$ and an element $t\in S$, let $E-t:=\\{s\in S:s+t\in
E\\}$ and $E+t:=\\{s+t:s\in S\\}$. The following definition summarizes
combinatorial notions of largeness that we will use, some of which are defined
above in the setting of abelian groups.

######  Definition 8.1.

Let $(S,+)$ be a countable cancellative abelian semigroup.

  * •

A set $E\subseteq S$ is _syndetic_ if there are finitely many elements
$t_{1},\dots,t_{k}\in S$ such that $\bigcup_{i=1}^{k}{(E-t_{i})}=S$.

  * •

A set $T\subseteq S$ is _thick_ if for any finite set $F\subseteq S$, there
exists $t\in S$ such that $F+t\subseteq T$.

  * •

A set $P\subseteq S$ is _piecewise syndetic_ if there is a syndetic set
$E\subseteq S$ and a thick set $T\subseteq S$ such that $P=E\cap T$.

  * •

A sequence $(F_{N})_{N\in\mathbb{N}}$ of finite subsets of $S$ is a _Følner
sequence_ if, for any $t\in S$,

$\displaystyle\frac{\left|(F_{N}+t)\triangle F_{N}\right|}{|F_{N}|}\to 0.$

  * •

The _lower Banach density_ of a set $E\subseteq S$ is the quantity

$\displaystyle d_{*}(E):=\inf{\left\\{\liminf_{N\to\infty}{\frac{\left|E\cap
F_{N}\right|}{|F_{N}|}}:(F_{N})_{N\in\mathbb{N}}~{}\text{is a F{\o}lner
sequence in}~{}S\right\\}}.$

  * •

The _upper Banach density_ of a set $E\subseteq S$ is the quantity

$\displaystyle d^{*}(E):=\sup{\left\\{\limsup_{N\to\infty}{\frac{\left|E\cap
F_{N}\right|}{|F_{N}|}}:(F_{N})_{N\in\mathbb{N}}~{}\text{is a F{\o}lner
sequence in}~{}S\right\\}}.$

The following is a standard characterization of syndetic and thick sets; see,
e.g. [BHM98, Section 2].

######  Proposition 8.2.

Let $(S,+)$ be a countable cancellative abelian semigroup.

  1. 1.

$E$ is syndetic if and only if $d_{*}(E)>0$ if and only if $E\cap
T\neq\emptyset$ for any thick set $T\subseteq S$;

  2. 2.

$T$ is thick if and only if $d^{*}(T)=1$ if and only if $T\cap E\neq\emptyset$
for any syndetic set $E\subseteq S$.

######  Lemma 8.3.

Let $(S,+)$ be a countable cancellative abelian semigroup. Then $S$ is thick
in $G_{S}$.

###### Proof.

Let $F\subseteq G_{S}$ be a finite set. Write $F=\\{s_{i}-t_{i}:1\leq i\leq
k\\}$, where $s_{i},t_{i}\in S$. Put $t=\sum_{i=1}^{k}{t_{i}}\in S$. Then

$\displaystyle F+t=\left\\{s_{i}+\sum_{j\neq i}{t_{j}}:1\leq i\leq
k\right\\}\subseteq S.$

∎

The fact that $S$ is thick in $G_{S}$ is closely related to the fact that any
Følner sequence in $S$ is also a Følner sequence in $G_{S}$, from which we
deduce the following density result:

######  Proposition 8.4.

Let $E\subseteq S$. Then $d_{S}^{*}(E)=d_{G_{S}}^{*}(E)$.

###### Proof.

To show the inequality $d_{G_{S}}^{*}(E)\geq d_{S}^{*}(E)$, it suffices to
show that any Følner sequence in $S$ is a Følner sequence in $G_{S}$. Let
$(F_{N})_{N\in\mathbb{N}}$ be a Følner sequence in $S$, and let $x\in G$. We
want to show

$\displaystyle\frac{\left|(F_{N}+x)\triangle F_{N}\right|}{|F_{N}|}\to 0.$

Write $x=s-t$ with $s,t\in S$. Then

$\displaystyle\frac{\left|(F_{N}+x)\triangle
F_{N}\right|}{|F_{N}|}=\frac{\left|(F_{N}+s)\triangle(F_{N}+t)\right|}{|F_{N}|}\leq\frac{\left|(F_{N}+s)\triangle
F_{N}\right|}{|F_{N}|}+\frac{\left|F_{N}\triangle(F_{N}+t)\right|}{|F_{N}|}\to
0.$

Hence, $(F_{N})_{N\in\mathbb{N}}$ is a Følner sequence in $G_{S}$ as claimed.

Now we show the reverse inequality $d_{S}^{*}(E)\geq d_{G_{S}}^{*}(E)$. If
$d_{G_{S}}^{*}(E)=0$, there is nothing to show, so assume
$d_{G_{S}}^{*}(E)>0$. Let $m$ be an invariant mean on $G_{S}$ such that
$m(E)=d_{G_{S}}^{*}(E)$. Put $c=m(S)\geq m(E)>0$. Then
$\widetilde{m}:=\frac{1}{c}m$ is an invariant mean on $S$. Moreover,
$\widetilde{m}(E)=\frac{1}{c}m(E)\geq m(E)=d_{G_{S}}^{*}(E)$. Therefore,
$d_{S}^{*}(E)\geq\widetilde{m}(E)\geq d_{G_{S}}^{*}(E)$. ∎

######  Lemma 8.5.

Suppose $E\subseteq G_{S}$ is syndetic in $G_{S}$. Then $E\cap S$ is syndetic
in $S$.

###### Proof.

Let $x_{1},\dots,x_{k}\in G_{S}$ such that
$\bigcup_{i=1}^{k}{(E-x_{i})}=G_{S}$. By Lemma 8.3, $S$ is thick, so we may
assume $x_{i}\in S$ for each $i=1,\dots,k$. We claim

$\displaystyle\bigcup_{i=1}^{k}{\left((E\cap S)-x_{i}\right)}\supseteq S.$

It suffices to check $(E\cap S)-x_{i}\supseteq(E-x_{i})\cap S$ for each
$i=1,\dots,k$. Suppose $y\in(E-x_{i})\cap S$, and let $t\in E$ such that
$t-x_{i}=y$. Then $t=y+x_{i}\in S+S\subseteq S$. Hence, $y\in(E\cap S)-x_{i}$
as desired. ∎

###  8.3. Extending main results to actions of cancellative abelian semigroups

Any homomorphism $\varphi:S\to S$ extends uniquely to a homomorphism
$\widetilde{\varphi}:G_{S}\to G_{S}$ via
$\widetilde{\varphi}\left(s-t\right)=\varphi(s)-\varphi(t)$. To extend our
Khintchine-type results to the semigroup setting, we need a condition on
$\varphi$ characterizing when $\widetilde{\varphi}(G_{S})$ has finite index in
$G_{S}$.

######  Proposition 8.6.

Let $(S,+)$ be a countable cancellative abelian semigroup. Let $\varphi:S\to
S$ be a homomorphism, and let $\widetilde{\varphi}:G_{S}\to G_{S}$ be the
group homomorphism $\widetilde{\varphi}(s-t):=\varphi(s)-\varphi(t)$. The
following are equivalent:

  1. (i)

$\varphi(S)$ is a piecewise syndetic subset of $S$;

  2. (ii)

$\widetilde{\varphi}(G_{S})$ has finite index in $G_{S}$.

###### Proof.

Let $T:=\varphi(S)$, and let $H:=\widetilde{\varphi}(G_{S})$. Note that
$H=T-T=G_{T}$.

(i)$\implies$(ii). Suppose $T$ is piecewise syndetic in $S$. Then
$d_{S}^{*}(T)>0$. Thus, by Proposition 8.4, $d_{G_{S}}^{*}(H)\geq
d_{G_{S}}^{*}(T)=d_{S}^{*}(T)>0$. But in the group $G_{S}$, we have the
identity

$\displaystyle d_{G_{S}}^{*}(H)=\frac{1}{[G_{S}:H]},$

so $[G_{S}:H]<\infty$.

(ii)$\implies$(i). Suppose $H$ has finite index in $G_{S}$. Then $H$ is a
syndetic subset of $G_{S}$, so $H\cap S$ is syndetic in $S$ by Lemma 8.5.
Moreover, by Lemma 8.3, $T$ is a thick subset of $H$. Let
$\widetilde{T}:=T\cup(S\setminus H)$ so that $T=\widetilde{T}\cap(H\cap S)$.
We claim that $\widetilde{T}$ is thick in $S$.

Let $F\subseteq S$ be a finite set. Put $F_{1}=F\cap H$ and $F_{2}=F\setminus
H$. Since $T$ is a thick subset of $H$, there exists $x\in H$ such that
$F_{1}+x\subseteq T$. Write $x=s-t$ with $s,t\in T\subseteq H\cap S$. Then
$F_{1}+s=F_{1}+x+t\subseteq T+t\subseteq T$. Now, since $s\in H\cap S$ and $H$
is a group, we have $F_{2}+s\subseteq S\setminus H$. Thus,
$F+s=(F_{1}+s)\cup(F_{2}+s)\subseteq T\cup(S\setminus H)=\widetilde{T}$.

This shows that $\widetilde{T}$ is a thick subset of $S$, so
$T=\widetilde{T}\cap(H\cap S)$ is piecewise syndetic in $S$. ∎

Now we can extend Theorems 1.11 and 1.13 to the semigroup setting:

######  Theorem 8.7.

Let $(S,+)$ be a countable cancellative abelian semigroup. Let
$\varphi,\psi:S\to S$ be homomorphisms. If at least two of the three
subsemigroups $\varphi(S)$, $\psi(S)$, and $(\varphi+\psi)(S)$ are piecewise
syndetic in $S$, then for any set $E\subseteq S$ with positive upper Banach
density $d_{S}^{*}(E)>0$ and any $\varepsilon>0$, the set

$\displaystyle\left\\{s\in
S:d_{S}^{*}\left(E\cap\left(E-\varphi(s)\right)\cap\left(E-(\varphi+\psi)(s)\right)\right)>d_{S}^{*}(E)^{3}-\varepsilon\right\\}$

is syndetic in $S$.

######  Remark 8.8.

We use the pair $\\{\varphi,\varphi+\psi\\}$ rather than $\\{\varphi,\psi\\}$
since the difference $\psi-\varphi$ is not necessarily defined as a map into
$S$.

###### Proof.

By Proposition 8.4, we have $\delta:=d_{G_{S}}^{*}(E)=d_{S}^{*}(E)>0$. Let
$\widetilde{\varphi}$ and $\widetilde{\psi}$ be the extensions of $\varphi$
and $\psi$ to $G_{S}$. By Proposition 8.6, at least two of the subgroups
$\widetilde{\varphi}(G_{S})$, $\widetilde{\psi}(G_{S})$, and
$\left(\widetilde{\varphi}+\widetilde{\psi}\right)(G_{S})$ have finite index
in $G_{S}$. Hence by Theorem 1.11, the set

$\displaystyle R:=\left\\{g\in
G:d_{G_{S}}^{*}\left(E\cap\left(E-\widetilde{\varphi}(g)\right)\cap\left(E-\left(\widetilde{\varphi}+\widetilde{\psi}\right)(g)\right)\right)>\delta^{3}-\varepsilon\right\\}$

is syndetic in $G_{S}$.

By Lemma 8.5, the set $R\cap S$ is syndetic $S$. But

$\displaystyle R\cap S=\left\\{s\in
S:d_{S}^{*}\left(E\cap\left(E-\varphi(s)\right)\cap\left(E-(\varphi+\psi)(s)\right)\right)>\delta^{3}-\varepsilon\right\\},$

so this completes the proof. ∎

######  Theorem 8.9.

Let $(S,+)$ be a countable cancellative abelian semigroup. Let
$a,b\in\mathbb{N}$. If at least one of the three subsemigroups $aS$, $bS$, or
$(a+b)S$ is piecewise syndetic in $S$, then for any set $E\subseteq S$ with
positive upper Banach density $d_{S}^{*}(E)>0$ and any $\varepsilon>0$, the
set

$\displaystyle\left\\{s\in
S:d_{S}^{*}\left(E\cap\left(E-as\right)\cap\left(E-(a+b)s\right)\right)>d_{S}^{*}(E)^{3}-\varepsilon\right\\}$

is syndetic in $S$.

###### Proof.

The proof is identical to the proof of Theorem 8.7, except one must use
Theorem 1.13 in place of Theorem 1.11. ∎

###  8.4. Two combinatorial questions

Applying Theorem 8.9 in the semigroup $(\mathbb{N},\cdot)$, for any
$E\subseteq\mathbb{N}$ with positive multiplicative upper Banach density
$d_{mult}^{*}(E)>0$, any $k\in\mathbb{N}$, and any $\varepsilon>0$, the set of
$m\in\mathbb{N}$ such that

$\displaystyle d^{*}_{mult}\left(E\cap E/m^{k}\cap
E/m^{k+1}\right)>d^{*}_{mult}(E)^{3}-\varepsilon$

is multiplicatively syndetic in $\mathbb{N}$. It is natural to ask if a
finitary variant of this result holds.

######  Question 8.10.

Let $p_{1},p_{2},\dots$ be an enumeration of the positive prime numbers. Let
$\delta,\varepsilon>0$, and let $k\in\mathbb{N}$. Does there exists
$N=N(k,\delta,\varepsilon)\in\mathbb{N}$ such that the following holds: for
any $n\geq N$ and any set $A\subseteq\left\\{p_{1}^{r_{1}}\dots
p_{n}^{r_{n}}:0\leq r_{i}\leq n\right\\}$ with $|A|\geq\delta n^{n}$, there
exists $y\in\mathbb{N}\setminus\\{1\\}$ such that

$\displaystyle\left|\left\\{x\in\mathbb{N}:\\{x,xy^{k},xy^{k+1}\\}\subseteq
A\right\\}\right|>\left(\delta^{3}-\varepsilon\right)n^{n}.$

Now we describe an application of Theorem 8.7. Let $p_{1},p_{2},\dots$ and
$q_{1},q_{2},\dots$ be enumerations of the positive prime numbers. The map
$\varphi:\mathbb{N}\to\mathbb{N}$ defined by
$\varphi\left(\prod_{i=1}^{n}{p_{i}^{r_{i}}}\right):=\prod_{i=1}^{n}{q_{i}^{r_{i}}}$
is an automorphism of the semigroup $(\mathbb{N},\cdot)$. Hence, by Theorem
8.7, if $E\subseteq\mathbb{N}$ has positive multiplicative upper Banach
density $d_{mult}^{*}(E)>0$ and $\varepsilon>0$, then there is a
multiplicatively syndetic set of numbers
$y=\prod_{i=1}^{n}{p_{i}^{r_{i}}}\in\mathbb{N}$ such that

(8.1) $\displaystyle
d_{mult}^{*}\left(\left\\{x\in\mathbb{N}:\left\\{x,x\prod_{i=1}^{n}{p_{i}^{r_{i}}},x\prod_{i=1}^{n}{q_{i}^{r_{i}}}\right\\}\subseteq
E\right\\}\right)>d_{mult}^{*}(E)^{3}-\varepsilon.$

The IP Szemerédi theorem of Furstenberg and Katznelson [FK85] implies that,
for any $k\in\mathbb{N}$ and any multiplicative automorphisms
$\varphi_{1},\dots,\varphi_{k}:\mathbb{N}\to\mathbb{N}$, the set of
$m\in\mathbb{N}$ such that

$\displaystyle d_{mult}^{*}\left(E\cap E/\varphi_{1}(m)\cap\dots\cap
E/\varphi_{k}(m)\right)>0$

is a multiplicative IP∗ set and hence multiplicatively syndetic. It is
therefore natural to ask if a large intersections variant holds for families
of more than two multiplicative automorphisms:

######  Question 8.11.

Let $p_{1},p_{2},\dots$ be the enumeration of the positive prime numbers in
increasing order. For each $j\in\mathbb{N}$, let $q_{j,1},q_{j,2},\dots$ be a
distinct enumeration of the positive prime numbers. For which $k\in\mathbb{N}$
does the following hold: for any $E\subseteq\mathbb{N}$ with
$d_{mult}^{*}(E)>0$ and any $\varepsilon>0$, there exists
$y=\prod_{i=1}^{n}{p_{i}^{r_{i}}}\in\mathbb{N}\setminus\\{1\\}$ such that

(8.2) $\displaystyle
d_{mult}^{*}\left(\left\\{x\in\mathbb{N}:\left\\{x,x\prod_{i=1}^{n}{q_{1,i}^{r_{i}}},x\prod_{i=1}^{n}{q_{2,i}^{r_{i}}},\dots,x\prod_{i=1}^{n}{q_{k,i}^{r_{i}}}\right\\}\subseteq
E\right\\}\right)>d_{mult}^{*}(E)^{k+1}-\varepsilon.$

Note that (8.2) holds for $k\leq 2$ (see (8.1) and the discussion above).

##  Appendix A Proof of Lemma 3.6

In this section we prove Lemma 3.6, restated here for the convenience of the
reader:

######  Lemma A.1 (Lemma 3.5).

Let $(X,\mathcal{X},\mu,(T_{g})_{g\in G})$ be a $G$-system and let $H\leq G$
be a subgroup of finite index. Then for every $k\geq 1$, one has
$\mathcal{Z}^{k}_{H}(X)=\mathcal{Z}^{k}_{G}(X)$.

We follow the arguments in [B06, Appendix A] and generalize them to arbitrary
countable abelian groups. We start with some background related to the
Host–Kra parallelepipeds construction.

######  Definition A.2.

Let $G$ be a countable abelian group, and let
$\textbf{X}=(X,\mathcal{X},\mu,(T_{g}))$ be a $G$-system. For every $k\geq 0$,
we define a $G$-system
$\textbf{X}_{G}^{[k]}=(X^{[k]},\mathcal{X}^{[k]},\mu^{[k]},(T^{[k]}_{g})_{g\in
G})$ inductively by setting $X_{G}^{[0]}=X$, and
$X_{G}^{[k+1]}=X_{G}^{[k]}\times_{\mathcal{I}(X_{G}^{[k]})}X_{G}^{[k]}$ where
$\mathcal{I}(X_{G}^{[k]})$ is the $\sigma$-algebra of $(T_{g}^{[k]})_{g\in
G}$-invariant functions.

Host and Kra proved the following result.

######  Theorem A.3 ([HK05], Proposition 4.7).

$\mathcal{Z}^{k}_{G}(X)$ is the minimal $\sigma$-algebra with the property
that $\mathcal{I}(X^{[k]})$ is a sub $\sigma$-algebra of
$(\mathcal{Z}^{k}_{G}(X))^{[k]}$.

Let $X=\bigcup_{\alpha\in J}X_{\alpha}$ be a partition of $X$ to $G$-invariant
sets. Then $X_{G}^{[k]}=\bigcup_{\alpha\in J}X_{\alpha}^{[k]}$,
$\mathcal{I}(X^{[k]})=\bigvee_{\alpha\in J}\mathcal{I}(X_{\alpha}^{[k]})$ and
$\mathcal{Z}^{k}_{G}(X)=\bigvee_{\alpha\in J}\mathcal{Z}^{k}_{G}(X_{\alpha})$.
Therefore, by the ergodic decomposition, it is enough to prove Lemma 3.6 in
the case where the $G$-action is ergodic.

The following lemma gives the easy inclusion in Lemma 3.6.

######  Lemma A.4.

In the setting of Lemma 3.6,
$\mathcal{Z}_{G}^{k}(X)\preceq\mathcal{Z}_{H}^{k}(X)$.

###### Proof.

The proof is immediate by Theorem A.3 and since any $(T_{g}^{[k]})_{g\in
G}$-invariant function is also a $(T_{h}^{[k]})_{h\in H}$-invariant function.
∎

We need the following observation.

######  Lemma A.5.

Let $G$ be a countable abelian group, let
$\textbf{X}=(X,\mathcal{X},\mu,(T_{g})_{g\in G})$ be an ergodic measure
preserving $G$-system, and let $H\leq G$ be a subgroup of finite index. Then
$\mathcal{I}_{H}(X)\preceq\mathcal{Z}_{G}(X)$.

###### Proof.

The group $G/H$ acts ergodically by unitary transformations on
$\mathcal{H}=L^{2}(X,\mathcal{I}_{H},\mu|_{\mathcal{I}_{H}})$. Since $G/H$ is
a finite abelian group, the unitary representation splits into a direct sum of
one-dimensional irreducible representations. In other words, $\mathcal{H}$ is
generated by eigenfunctions of the action of $G/H$, which are measurable with
respect to $\mathcal{Z}_{G}(X)$. This completes the proof. ∎

Now we prove the $k=1$ case of Lemma 3.6 under the additional assumption that
the action of $H$ is ergodic.

######  Lemma A.6.

Let $G$ be countable abelian groups, and let $H\leq G$ be a finite index
subgroup. Let $\textbf{X}=(X,\mathcal{X},\mu,(T_{g})_{g\in G})$ be an ergodic
$G$-system, and suppose the action of $H$ is ergodic. Then
$\mathcal{Z}_{H}(X)=\mathcal{Z}_{G}(X)$.

###### Proof.

The group $G/H$ is finite, and therefore it is a direct product of finite
cyclic groups. In particular, we can find $d\in\mathbb{N}$ and a sequence of
subgroups $H_{0}=H\leq H_{1}\leq\dots\leq H_{d}\leq G$ such that $G/H_{d}$ and
$H_{i}/H_{i-1}$, $1\leq i\leq d$, are cyclic groups of prime order. Using a
proof by induction on $d$, we may assume without loss of generality that $G/H$
is cyclic and of prime order. Let $g_{0}\in G$ be a representative of a
generator of $G/H$ and $l:=[G:H]$ be a prime number. By the ergodicity of $H$,
the $\sigma$-algebra $\mathcal{Z}_{H}(X)$ is generated by $H$-eigenfunctions.
Hence, it is enough to show that every $H$-eigenfunction $f$ is a linear
combination of $G$-eigenfunctions. Let $\lambda:H\rightarrow S^{1}$ be the
eigenvalue of $f$ and observe that for any $l$-th root $\omega\in S^{1}$ of
$\lambda(lg_{0})$ the function

$f+\omega\cdot T_{g_{0}}f+...+\omega^{l-1}\cdot T_{(l-1)g_{0}}f$

is a $G$-eigenfunction. Now since

$f=\sum_{\omega\in S^{1}~{}:~{}\omega^{l}=\lambda(lg_{0})}f+\omega\cdot
T_{g_{0}}f+...+\omega^{l-1}\cdot T_{(l-1)g_{0}}f,$

$f$ is measurable with respect to $\mathcal{Z}_{G}(X)$ and this completes the
proof. ∎

Let $G$ be a countable abelian group, and let
$\textbf{X}=(X,\mathcal{X},\mu,(T_{g})_{g\in G})$ be a $G$-system. If the
system X is ergodic, it follows from the definition that $X_{G}^{[1]}$ is the
Cartesian product of $X$ with itself, and the measure is the product measure.
As a consequence of Lemma A.6, we have:

######  Lemma A.7.

If the action of $H$ on $X$ is ergodic, then

$\mathcal{I}(X_{H}^{[1]})=\mathcal{I}(X_{G}^{[1]}).$

###### Proof.

The inclusion $\mathcal{I}(X_{G}^{[1]})\preceq\mathcal{I}(X_{H}^{[1]}))$ is
trivial. Now let $f:X\times X\rightarrow\mathbb{C}$ be a $(T_{h}\times
T_{h})_{h\in H}$ invariant function. By Lemma A.6, we can find an orthonormal
basis of $G$-eigenfunctions $\\{f_{i}\\}_{i\in\mathbb{N}}$ for
$\mathcal{Z}_{H}(X)$. By Lemma 4.6, there exist constants
$a_{i,j}\in\mathbb{C}$ for all $i,j\in\mathbb{N}$ such that

$f(x,y)=\sum_{i=1}^{\infty}a_{i,j}f_{i}(x)\overline{f_{j}}(y).$

Applying the $H$-action and using the uniqueness of the decomposition, we see
that $a_{i,j}=0$ unless $i=j$. In particular $f$ is spanned by the
$G$-invariant functions $f_{i}\otimes\overline{f_{i}}$. Thus, $f$ is
measurable with respect to $\mathcal{I}(X_{G}^{2})$ and the claim follows. ∎

We use Lemma A.7 to prove the following:

######  Proposition A.8.

If the action of $H$ on $X$ is ergodic, then for $k\geq 0$, one has

$\mathcal{I}(X_{H}^{[k]})=\mathcal{I}(X_{G}^{[k]})\qquad\text{and}\qquad\mu_{G}^{[k]}=\mu_{H}^{[k]}.$

###### Proof.

We prove the claim by induction on $k$. The case $k=0$ is trivial.

Assume that for some $k\geq 0$,
$\mathcal{I}(X_{H}^{[k]})=\mathcal{I}(X_{G}^{[k]})$ and
$\mu_{G}^{[k]}=\mu_{H}^{[k]}$. It is immediate that

$\mu_{G}^{[k+1]}=\mu_{G}^{[k]}\times_{\mathcal{I}(X_{G}^{[k]})}\mu_{G}^{[k]}=\mu_{H}^{[k]}\times_{\mathcal{I}(X_{H}^{[k]})}\mu_{H}^{[k]}=\mu_{H}^{[k+1]}.$

By the ergodic decomposition theorem, applied with respect to the
$\sigma$-algebra $\mathcal{I}(X_{G}^{[k]})$ we can find a partition
$X_{G}^{[k]}=\bigcup_{\alpha\in J}X_{\alpha}$ of $X_{G}^{[k]}$ to
$(T_{g}^{[k]})_{g\in G}$ invariant sets. Let $S_{g}^{\alpha}$ be the
restriction of $T_{g}^{[k]}$ to the set $X_{\alpha}$. By the induction
hypothesis the action of $(S_{h}^{\alpha})_{h\in H}$ on $X_{\alpha}$ is
ergodic. Hence, by Lemma A.7, we have

$\mathcal{I}(X_{H}^{[k+1]})=\bigcup_{\alpha\in
J}\mathcal{I}_{H}(X_{\alpha}^{[1]}))=\bigcup_{\alpha\in
J}\mathcal{I}_{G}(X_{\alpha}^{[1]})=\mathcal{I}(X_{G}^{[k+1}),$

as required. ∎

Proposition A.8 establishes Lemma 3.6 in the case where the action of $H$ is
ergodic. Now we assume that the $H$-action is non-ergodic. As in the proof of
Lemma A.6, we may assume without loss of generality that $G/H$ is cyclic of
order $l$ for some prime $l$. In particular, there exists a partition
$X=\bigcup_{i\in\mathbb{Z}/l\mathbb{Z}}X_{i}$ into $H$-invariant sets and some
$g_{0}\in G$ such that $T_{g_{0}}X_{i}=X_{i+1}$, $i\in\mathbb{Z}/l\mathbb{Z}$.

We need the following technical lemma.

######  Lemma A.9.

Let $G$ be a countable abelian group, and let
$\textbf{Y}=(Y,\mathcal{Y},\nu,(T_{g})_{g\in G})$ be an ergodic $G$-system.
Suppose that there exists some $g_{0}\in G$ and $H$-invariant subsets $Y_{i}$
such that $Y=\bigcup_{i\in\mathbb{Z}/l\mathbb{Z}}Y_{i}$, and
$T_{g_{0}}Y_{i}=Y_{i+1}$ for $i\in\mathbb{Z}/l\mathbb{Z}$. Then,
$Y\times_{\mathcal{I}_{G}(Y)}Y=\bigcup_{i,j\in\mathbb{Z}/l\mathbb{Z}}Y_{i,j}$
where $Y_{i,i}=Y_{i}\times_{\mathcal{I}_{H}(Y_{i})}Y_{i}$ and
$T_{sg_{0}}\times T_{tg_{0}}$ is an isomorphism between $Y_{i,i}$ and
$Y_{i+s,i+t}$, $i\in\mathbb{Z}/l\mathbb{Z}$.

###### Proof.

Let $A\in\mathcal{I}_{G}(Y)$ be a measurable $G$-invariant subset of $Y$. For
each $0\leq i\leq l-1$, $A_{i}=A\cap Y_{i}$ is an $H$-invariant set. In
particular, $A_{0}$ is $H$-invariant and $A_{i}=T_{ig_{0}}A_{0}$. We deduce
that the mapping $A\mapsto A\cap Y_{0}$ is an isomorphism between
$\mathcal{I}_{G}(Y)$ and $\mathcal{I}_{H}(Y_{0})$. Using the ergodic
decomposition we can find a partition

$Y_{0}=\bigcup_{\alpha\in I}Y_{0,\alpha}$

of $Y_{0}$ to $H$-invariant sets. For every $\alpha\in I$, and $i\not=0$, let
$Y_{i,\alpha}=T_{ig_{0}}Y_{0,\alpha}$ and
$Y_{\alpha}=\bigcup_{i\in\mathbb{Z}/l\mathbb{Z}}Y_{i,\alpha}$. Then,
$Y=\bigcup_{\alpha\in I}Y_{\alpha}$ is the ergodic decomposition of $Y$ with
respect to the factor $\mathcal{I}_{G}(Y)$. Thus, if we let
$Y_{i,j}=\bigcup_{\alpha\in I}Y_{i,\alpha}\times Y_{j,\alpha}$ we have,

$Y_{G}^{[1]}=\bigcup_{\alpha\in
I}(Y_{\alpha}\times_{\mathcal{I}_{G}(Y_{\alpha})}Y_{\alpha})=\bigcup_{\alpha\in
I}\bigcup_{i,j\in\mathbb{Z}/l\mathbb{Z}}(Y_{i,\alpha}\times
Y_{j,\alpha})=\bigcup_{i,j\in\mathbb{Z}/l\mathbb{Z}}\bigcup_{\alpha\in I}\
(Y_{i,\alpha}\times
Y_{j,\alpha})=\bigcup_{i,j\in\mathbb{Z}/l\mathbb{Z}}Y_{i,j}.$

In particular, $Y_{i,i}=\bigcup_{\alpha\in I}(Y_{i,\alpha}\times
Y_{i,\alpha})=Y_{i}\times Y_{i}$, as required. ∎

Recall that $G=\bigcup_{i=0}^{l-1}ig_{0}+H$. It follows from Lemma A.9 that
for $i,j\in\mathbb{Z}/l\mathbb{Z}$,

$(T_{g_{0}}\times
T_{g_{0}})(Y_{i}\times_{\mathcal{I}_{H}(Y)}Y_{j})=Y_{i+1,j+1}.$

Therefore, the subsets $V_{i}=\bigcup_{j\in\mathbb{Z}/l\mathbb{Z}}Y_{j,j+i}$,
$i\in\mathbb{Z}/l\mathbb{Z}$ form a partition of
$Y\times_{\mathcal{I}_{G}(Y)}Y$ into $(T_{g}\times T_{g})_{g\in G}$-invariant
sets. Furthermore, $\text{Id}\times T_{ig_{0}}$ is an isomorphism between
$V_{0}$ and $V_{i}$.

We use Lemma A.9 to show the following:

######  Lemma A.10.

Let $\textbf{X}=(X,\mathcal{X},\mu,(T_{g})_{g\in G})$ be an ergodic
$G$-system. Let $X=\bigcup_{i\in\mathbb{Z}/l\mathbb{Z}}X_{i}$ be a partition
into $H$-invariant sets, and let $g_{0}\in G$ be as above. Then for any $k\geq
0$, there exists a partition
$X_{G}^{[k]}=\bigcup_{j\in(\mathbb{Z}/l\mathbb{Z})^{k}}W_{j}$, into
$(T_{g}^{[k]})_{g\in G}$-invariant sets, such that
$W_{0}=\bigcup_{i\in\mathbb{Z}/l\mathbb{Z}}(X_{i})_{H}^{[k]}$ and
$T_{g_{0}}^{[k]}\left((X_{i})_{H}^{[k]}\right)=(X_{i+1})_{H}^{[k]}$.
Furthermore, for every $j\in(\mathbb{Z}/l\mathbb{Z})^{k}$, there exists an
isomorphism of measure spaces $\tau_{j}:W_{0}\rightarrow W_{j}$, which in
every coordinate of $X^{[k]}$ is a power of $T_{g_{0}}$.

###### Proof.

We induct on $k$. The case $k=0$ is trivial.

Assume that the claim holds for some $k\geq 0$. Then

$X_{G}^{[k+1]}=X_{G}^{[k]}\times_{\mathcal{I}(X_{G}^{[k]})}X_{G}^{[k]}=\bigcup_{j\in(\mathbb{Z}/l\mathbb{Z})^{k}}(W_{j}\times_{\mathcal{I}(W_{j})}W_{j}).$

Fix $j\in(\mathbb{Z}/l\mathbb{Z})^{k}$. Since the isomorphism
$\tau_{j}:W_{0}\rightarrow W_{j}$ commutes with $(T_{g}^{[k]})_{g\in G}$, it
induces an isomorphism
$\tau_{j}\times\tau_{j}:W_{0}\times_{\mathcal{I}(W_{0})}W_{0}\rightarrow
W_{j}\times_{\mathcal{I}(W_{j})}W_{j}$. By assumption
$W_{0}=\bigcup_{i\in\mathbb{Z}/l\mathbb{Z}}(X_{i})_{H}^{[k]}$ and by Lemma
A.9, $W_{0}\times_{\mathcal{I}(W_{0})}W_{0}$ can be partitioned into
$(T_{g}^{[k+1]})_{g\in G}$-invariant sets
$\\{V_{i}\\}_{i\in\mathbb{Z}/l\mathbb{Z}}$ such that

$V_{0}=\bigcup_{i\in\mathbb{Z}/l\mathbb{Z}}\left((X_{i})_{H}^{[k]}\times_{\mathcal{I}\left((X_{i})_{H}^{[k]}\right)}(X_{i})_{H}^{[k]}\right)=\bigcup_{i\in\mathbb{Z}/l\mathbb{Z}}(X_{i})_{H}^{[k+1]}.$

Moreover, $V_{0}$ is isomorphic to $V_{j}$ via an isomorphism whose
projections are powers of $T_{g_{0}}^{[k]}$. Since $W_{0}$ is isomorphic to
$W_{j}$, this completes the proof. ∎

We recall that it suffices to establish the proof of Lemma 3.6 in the case
where the $G$-action is ergodic and $G/H$ is a cyclic group of order $l$ for
some $l>0$. As before, we find a partition
$X=\bigcup_{i\in\mathbb{Z}/l\mathbb{Z}}X_{i}$ of $X$ into $H$-invariant sets
and some $g_{0}\in G$ such that $T_{g_{0}}(X_{i})=X_{i+1}$ for
$i\in\mathbb{Z}/l\mathbb{Z}$.

###### Proof of Lemma 3.6.

Let $k\geq 0$, and let $\\{W_{i}\\}_{i\in(\mathbb{Z}/l\mathbb{Z})^{k}}$ be as
in Lemma A.10. Since $X_{0},...,X_{l-1}$ are disjoint $(T_{h})_{h\in
H}$-invariant subsets of $X$, we have
$\mathcal{I}(X_{H}^{[k]})=\prod_{i\in\mathbb{Z}/l\mathbb{Z}}\mathcal{I}\left((X_{i})_{H}^{[k]}\right)$
and $Z^{k}_{H}(X)=\prod_{i\in\mathbb{Z}/l\mathbb{Z}}Z^{k}_{H}(X_{i})$. Let $B$
be a $(T_{h}^{[k]})_{h\in H}$-invariant subset of $(X_{i})_{H}^{[k]}$. For
every $j\in\mathbb{Z}/l\mathbb{Z}$, let $A_{j}=(T_{(j-i)g_{0}}^{[k]})(B)$ and
$A=\bigcup_{j\in\mathbb{Z}/l\mathbb{Z}}A_{j}$. By definition $A\subseteq
W_{0}$ is a $(T_{g}^{[k]})_{g\in G}$-invariant set. Therefore, by Theorem A.3,
$A\in\left(\mathcal{Z}^{k}_{G}(X)\right)^{[k]}$. Since $X_{i}$ is
$(T_{h}^{[k]})$-invariant, by Lemma A.5, $X_{i}\in\mathcal{Z}^{1}_{G}(X)$.
Therefore, $B=A_{i}=A\cap\left(X_{i}\right)_{H}^{[k]}$ is an element of
$\left(\mathcal{Z}^{k}_{G}(X)\right)^{[k]}$. Since $B$ is arbitrary, and this
holds for all $i\in\mathbb{Z}/l\mathbb{Z}$, we deduce that
$\mathcal{I}(X_{H}^{[k]})\preceq\mathcal{Z}^{k}_{G}(X)$. By Theorem A.3, we
have $\mathcal{Z}^{k}_{H}(X)\preceq\mathcal{Z}_{G}^{k}(X)$. Lemma A.4 provides
the other inclusion, and this completes the proof. ∎

## References

  * [AB21] E. Ackelsberg and V. Bergelson.  Popular differences for polynomial patterns in rings of integers.  arXiv:2107.07626.
  * [ABB21] E. Ackelsberg, V. Bergelson, and A. Best.  Multiple recurrence and large intersections for abelian group actions.  _Discrete Anal._ 2021:18, 91 pp.
  * [AWM60] F. V. Atkinson, G. A. Watterson and P. A. P. Moran.  A matrix inequality.  _Quart. J. Math. Oxford Ser. (2)_ , 11:137–140, 1960.
  * [Au16] T. Austin.  Non-conventional ergodic averages for several commuting actions of an amenable group.  _J. Anal. Math._ , 130:243–274, 2016.
  * [Beh46] F. A. Behrend.  On sets of integers which contain no three terms in arithmetical progression.  _Proc. Nat. Acad. Sci. U.S.A._ , 32:331–332, 1946.
  * [B87] V. Bergelson.  Weakly mixing PET.  _Ergodic Theory Dynam. Systems_ , 7:337–349, 1987.
  * [B06] V. Bergelson.  Combinatorial and Diophantine applications of ergodic theory. Appendix A by A. Leibman and Appendix B by A. Quas and M. Wierdl.  In _Handbook of dynamical systems._ , Vol. 1B, 745–869, Elsevier B. V., Amsterdam 2006.
  * [BF21] V. Bergelson and A. Ferré Moragues.  An ergodic correspondence principle, invariant means and applications.  _Israel J. Math._ , to appear. arXiv:2003:03029.
  * [BHM98] V. Bergelson, N. Hindman, and R. McCutcheon.  Notions of size and combinatorial properties of quotient sets in semigroups.  In _Proceedings of the 1998 Topology and Dynamics Conference (Fairfax, VA)_ , Topology Proc., 23:23–60, 1998.
  * [BHK05] V. Bergelson, B. Host and B. Kra.  Multiple recurrence and nilsequences. With an appendix by Imre Ruzsa.  _Invent. Math._ , 160(2):261–303, 2005.
  * [BL15] V. Bergelson and A. Leibman.  Cubic averages and large intersections.  In _Recent Trends in Ergodic Theory and Dynamical Systems_ , volume 631 of Contemp. Math., 5–19, Amer. Math. Soc., Providence, RI, 2015.
  * [BLL08] V. Bergelson, A. Leibman, and E. Lesigne.  Intersective polynomials and the polynomial Szemerédi theorem.  _Adv. Math._ , 219(1):369–388, 2008.
  * [BTZ10] V. Bergelson, T. Tao, and T. Ziegler.  An inverse theorem for the uniformity seminorms associated with the action of $\mathbb{F}_{p}^{\infty}$ _Geom. Funct. Anal._ , 19(6):1539–1596, 2010.
  * [Ber21] A. Berger.  Popular differences for corners in Abelian groups.  _Math. Proc. Cambridge Philos. Soc_ , 171(1):207–225, 2021.
  * [BSST21] A. Berger, A. Sah, M. Sawhney, and J. Tidor.  Popular differences for matrix patterns.  _Trans. Amer. Math. Soc._ , to appear. arXiv:2102.01684.
  * [C11] Q. Chu.  Multiple recurrence for two commuting transformations.  _Ergodic Theory Dynam. Systems_ , 31(3):771–792, 2011.
  * [CFH11] Q. Chu, N. Frantzikinakis, and B. Host.  Commuting averages with polynomial iterates of distinct degrees.  _Proc. Lond. Math. Soc. (3)_ 102(5):801–842, 2011.
  * [DLMS21] S. Donoso, A. Le, J. Moreira, and W. Sun.  Optimal lower bounds for multiple recurrence.  _Ergodic Theory Dynam. Systems_ , 41:379–407, 2021.
  * [DS18] S. Donoso and W. Sun.  Quantitative multiple recurrence for two and three transformations.  _Israel J. Math._ , 226(1):71–85, 2018.
  * [FSSSZ20] J. Fox, A. Sah, M. Sawhney, D. Stoner, and Y. Zhao.  Triforce and corners.  _Math. Proc. Cambridge Philos. Soc._ , 169(1):209–223, 2020.
  * [F08] N. Frantzikinakis.  Multiple ergodic averages for three polynomials and applications.  _Trans. Amer. Math. Soc._ , 360(10):5435–5475, 2008.
  * [FH18] N. Frantzikinakis, B. Host.  Weighted multiple ergodic averages and correlation sequences.  _Ergodic Theory Dynam. Systems_ , 38(1):81–142, 2018.
  * [FK85] H. Furstenberg and Y. Katznelson.  An ergodic Szemerédi theorem for IP-systems and combinatorial theory.  _J. Analyse Math._ , 45:117–168, 1985.
  * [FW96] H.Furstenberg and B. Weiss.  A Mean ergodic theorem for $\frac{1}{N}\sum_{n=1}^{N}f(T^{n}(x))g(T^{n^{2}}(x))$ In _Convergence in Ergodic Theory and Probability, Columbus, OH 1993 (Bergelson, March, and Rosenblatt, eds.)_ , Ohio State Univ. Math. Res. Inst. Publ. 5, de Gruyter, Berlin (1996), 193-227.
  * [G01] T. Gowers.  A new proof of Szemeredi’s theorem.  _Geom. Func. Anal._ , 11:465–588, 2001.
  * [HK05] B. Host and B. Kra.  Nonconventional ergodic averages and nilmanifolds.  _Ann. Math._ , 161:397–488, 2005.
  * [JST21] A. Jamneshan, O. Shalom, and T. Tao.  The structure of arbitrary Conze–Lesigne systems.  arXiv:2112.02056.
  * [Kh35] A. Khintchine.  Eine Verschärfung des Poincaréschen “Wiederkehrsatzes”.  _Compositio Math._ , 1:177–179, 1935.
  * [M21] M. Mandache.  A variant of the Corners theorem.  _Math. Proc. Cambridge Philos. Soc._ , 171(3):607–621, 2021.
  * [MS80] C. Moore and K. Schmidt.  Coboundaries and homomorphisms for non-singular actions and a problem of H. Helson.  _Proc. London Math. Soc. (3)_ , 40:443–475, 1980.
  * [R90] W. Rudin.  _Fourier Analysis on Groups_ ,  Wiley Classics Library, John Wiley & Sons, Inc., New York, 1990.
  * [SSZ21] A. Sah, M. Sawhney, and Y. Zhao.  Patterns without a popular difference.  _Discrete Anal._ , 2021:8, 30 pp.
  * [S21] O. Shalom.  Multiple ergodic averages in abelian groups and Khintchine type recurrence.  _Trans. Amer. Math. Soc._ , to appear. arXiv:2102.07273.
  * [Sri98] S. M. Srivastava.  _A Course on Borel Sets_ , volume 180 of Graduate Texts in Mathematics, Springer-Verlag, New York, 1998.
  * [TZ16] T. Tao and T. Ziegler.  Concatenation theorems for anti-Gowers-uniform functions and Host–Kra characteristic factors.  _Discrete Anal._ , 2016:13, 60 pp.
  * [Zie07] T. Ziegler.  Universal characteristic factors and Furstenberg averages.  _J. Amer. Math. Soc._ , 20:53–97, 2007.
  * [Zim76] R. Zimmer.  Extensions of ergodic group actions.  _Illinois J. Math._ 20:373–409, 1976.
  * [Z-K16] P. Zorin-Kranich.  Norm convergence of multiple ergodic averages on amenable groups.  _J. Anal. Math._ , 130:219–241, 2016.