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ON DIFFERENCE SETS OF DENSE SUBSETS OF $\mathbb {Z}^2$

Part of: Sequences and sets Designs and configurations Extremal combinatorics

Published online by Cambridge University Press:  24 September 2026

SAYAN GOSWAMI Open the ORCID record for SAYAN GOSWAMI [Opens in a new window]
SAYAN GOSWAMI*
Affiliation:
Ramakrishna Mission Vivekananda Educational and Research Institute, Belur, Howrah, 711202, India
*
e-mail: sayan20math@gmail.com
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Abstract

We study the structure of the difference set $E - E$ for subsets $E \subseteq \mathbb {Z}^2$ of positive upper Banach density. Fish [‘On product of difference sets for sets of positive density’, Proc. Amer. Math. Soc. 146 (2018), 3449–3453] asked in Problem 2 whether, for every such set E, there exists a nonzero integer k such that $k \cdot \mathbb {Z} \subseteq \{ xy : (x,y) \in E - E \}$. Although this question remains open, we establish a relatively weaker form. Specifically, we prove that if $\langle a_j\rangle _{j=1}^m$ is any finite sequence in $\mathbb {N},$ then there exist infinitely many integers $k \in \mathbb {Z}$ and infinitely many sequences $\langle x_n \rangle _{n \in \mathbb {N}}$ in $\mathbb {Z}$ such that $k \cdot MT(\langle a_j \rangle _{j=1}^m, \langle x_n\rangle _{n}) \subseteq \{ xy : (x,y) \in E - E \}$, where $MT(\langle a_j \rangle _{j=1}^m, \langle x_n\rangle _{n})$ denotes the Milliken–Taylor configuration generated by the sequences $\langle a_j\rangle _{j=1}^m$ and $\langle x_n \rangle _{n \in \mathbb {N}}$.

Keywords

sum-product estimatesdifference setproduct sets

MSC classification

Primary: 11B13: Additive bases, including sumsets
Secondary: 05B10: Difference sets (number-theoretic, group-theoretic, etc.) 05D10: Ramsey theory 11B05: Density, gaps, topology

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Type
Research Article
Information
Bulletin of the Australian Mathematical Society , First View , pp. 1 - 7
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Copyright
© The Author(s), 2026. Published by Cambridge University Press on behalf of Australian Mathematical Publishing Association Inc.

1 Introduction

Estimating the sizes of sumsets and product sets of subsets of $\mathbb {Z}$ double struck upper Z is a central topic in additive combinatorics. For any two sets $A,B \subset \mathbb {Z}$ upper A comma upper B subset of double struck upper Z (or $\mathbb {R}$ double struck upper R ), we define their sumset and product set by

$$ \begin{align*} A+B=\{a+b : a\in A,\ b\in B\} \quad \text{and} \quad A\cdot B=\{ab : a\in A,\ b\in B\}. \end{align*} $$ upper A plus upper B equals left brace a plus b colon a element of upper A comma b element of upper B right brace and upper A dot upper B equals left brace a b colon a element of upper A comma b element of upper B right brace period

One of the most fundamental open problems in sum–product theory is the following conjecture of Erdős and Szemerédi.

Conjecture 1.1 (Erdős and Szemerédi [Reference Erdős, Szemerédi, Erdős, Alpár, Halász and Sárközy4]).

If $A \subset \mathbb {Z}$ upper A subset of double struck upper Z is a finite set, then for every $\epsilon>0$ epsilon greater than 0 ,

$$ \begin{align*} |A+A| + |A\cdot A| \gg |A|^{2-\epsilon}. \end{align*} $$ StartAbsoluteValue upper A plus upper A EndAbsoluteValue plus StartAbsoluteValue upper A dot upper A EndAbsoluteValue much greater than StartAbsoluteValue upper A EndAbsoluteValue Superscript 2 minus epsilon Baseline period

If one replaces the finite set A by an infinite one, then an immediate question arises about the structures which are preserved under combinations of such sum–product sets. For various questions on sum, difference and product sets, we refer to the article [Reference Ruzsa9]. In [Reference Bjőrklund and Fish2, Reference Fish5], Bjőrklund and Fish initiated an infinitary version of these types of questions. For some recent developments, we refer to [Reference Bulinski and Fish3].

Definition 1.2 (Upper Banach density in $\mathbb {Z}^2$ double struck upper Z squared ).

Let $A \subseteq \mathbb {Z}^2$ upper A subset of or equal to double struck upper Z squared . The upper Banach density of A is defined by

$$ \begin{align*} d^\star(A) := \lim_{N \to \infty} \sup_{(m,n)\in \mathbb{Z}^2} \frac{ |A \cap ( (m,n) + [-N,N]^2 )| }{ (2N+1)^2 }. \end{align*} $$ d Superscript star Baseline left parenthesis upper A right parenthesis colon equals limit Underscript upper N right arrow infinity Endscripts sup Underscript left parenthesis m comma n right parenthesis element of double struck upper Z squared Endscripts StartFraction StartAbsoluteValue upper A intersection left parenthesis left parenthesis m comma n right parenthesis plus left bracket negative upper N comma upper N right bracket squared right parenthesis EndAbsoluteValue Over left parenthesis 2 upper N plus 1 right parenthesis squared EndFraction period

In [Reference Fish5], Fish asked the following question.

Problem 1.3 [Reference Fish5, Problem 2].

Is it true that for every set $E \subset \mathbb {Z}^2$ upper E subset of double struck upper Z squared of positive upper Banach density, there exists a constant $k_0$ k 0 , depending only on $d^\star (E)$ d Superscript star Baseline left parenthesis upper E right parenthesis , such that ${ k\,\mathbb {Z} \subseteq \{xy : (x,y) \in E - E \} }$ k double struck upper Z subset of or equal to StartSet x y colon left parenthesis x comma y right parenthesis element of upper E minus upper E EndSet for some integer $k \le k_0$ k less than or equals k 0 ?

Although this problem is still open, we prove a relatively weaker version. Before we state our result, let us recall the notion of IP sets. Let $\mathcal {P}_{f}(\mathbb {N})$ script upper P Subscript f Baseline left parenthesis double struck upper N right parenthesis be the collection of nonempty finite subsets of $\mathbb {N}.$ double struck upper N period In any commutative semigroup $(S,+),$ left parenthesis upper S comma plus right parenthesis comma a set A is said to be an IP set if there exists a sequence $\langle x_n\rangle _n$ left angle bracket x Subscript n Baseline right angle bracket Subscript n such that

$$ \begin{align*}A= {FS}(\langle x_{n}\rangle_{n\in\mathbb{N}})=\bigg\{ \sum_{t\in H}x_{t}:H\in\mathcal{P}_{f}(\mathbb{N})\bigg\}.\end{align*} $$ upper A equals upper F upper S left parenthesis left angle bracket x Subscript n Baseline right angle bracket Subscript n element of double struck upper N Baseline right parenthesis equals StartSet sigma summation Underscript t element of upper H Endscripts x Subscript t Baseline colon upper H element of script upper P Subscript f Baseline left parenthesis double struck upper N right parenthesis EndSet period

Over the semigroup $(\mathbb {N},\cdot ),$ left parenthesis double struck upper N comma dot right parenthesis comma we denote it by $FP(\langle x_{n}\rangle _{n\in \mathbb {N}}).$ upper F upper P left parenthesis left angle bracket x Subscript n Baseline right angle bracket Subscript n element of double struck upper N Baseline right parenthesis period The IP sets have significant importance in Ramsey theory; for example, it is known that they are partition regular (see [Reference Hindman6]).

An extension of Hindman’s finite sum theorem was studied independently by Milliken and Taylor in [Reference Milliken8, Reference Taylor10]. To address their results, we need the notion of a Milliken–Taylor system.

Definition 1.4 (Milliken–Taylor system).

Let $m\in \mathbb {N}$ m element of double struck upper N , and let $\langle a_j\rangle _{j=1}^m$ left angle bracket a Subscript j Baseline right angle bracket Subscript j equals 1 Superscript m and $\langle x_n\rangle _{n}$ left angle bracket x Subscript n Baseline right angle bracket Subscript n be sequences in $\mathbb {N}.$ double struck upper N period The Milliken–Taylor system determined by $\langle a_j\rangle _{j=1}^m$ left angle bracket a Subscript j Baseline right angle bracket Subscript j equals 1 Superscript m and $\langle x_n\rangle _{n}$ left angle bracket x Subscript n Baseline right angle bracket Subscript n is

$$ \begin{align*}MT(\langle a_j \rangle_{j=1}^m, \langle x_n\rangle_{n})=\bigg\lbrace \sum_{j=1}^m a_j \cdot \sum_{t\in F_j}x_t:F_1<F_2<\cdots <F_m \text{ for all } F_i\in \mathcal{P}_{f}(\mathbb{N})\bigg\rbrace.\end{align*} $$ upper M upper T left parenthesis left angle bracket a Subscript j Baseline right angle bracket Subscript j equals 1 Superscript m Baseline comma left angle bracket x Subscript n Baseline right angle bracket Subscript n Baseline right parenthesis equals StartSet sigma summation Underscript j equals 1 Overscript m Endscripts a Subscript j Baseline dot sigma summation Underscript t element of upper F Subscript j Baseline Endscripts x Subscript t Baseline colon upper F 1 less than upper F 2 less than midline horizontal ellipsis less than upper F Subscript m Baseline for all upper F Subscript i Baseline element of script upper P Subscript f Baseline left parenthesis double struck upper N right parenthesis EndSet period

Clearly, for any sequence $\langle x_n\rangle _{n},$ left angle bracket x Subscript n Baseline right angle bracket Subscript n Baseline comma we have ${FS}(\langle x_n\rangle _{n})=MT(\{1\}, \langle x_n\rangle _{n}).$ upper F upper S left parenthesis left angle bracket x Subscript n Baseline right angle bracket Subscript n Baseline right parenthesis equals upper M upper T left parenthesis StartSet 1 EndSet comma left angle bracket x Subscript n Baseline right angle bracket Subscript n Baseline right parenthesis period The following theorem is a more general result than the finite sum theorem (the notation is explained in Section 2.2). Although in the original text the result was described over the set of positive integers, the technique for $\mathbb {Z}$ double struck upper Z is similar.

Theorem 1.5 [Reference Hindman and Strauss7, Theorem 17.31].

Let $\langle a_j\rangle _{j=1}^m$ left angle bracket a Subscript j Baseline right angle bracket Subscript j equals 1 Superscript m be sequences in $\mathbb {N}.$ double struck upper N period Then, for any $p\in E(\beta \mathbb {Z},+)$ p element of upper E left parenthesis beta double struck upper Z comma plus right parenthesis and any $A\in a_1p+\cdots +a_mp,$ upper A element of a 1 p plus midline horizontal ellipsis plus a Subscript m Baseline p comma there exists a sequence $\langle x_n\rangle _{n}$ left angle bracket x Subscript n Baseline right angle bracket Subscript n such that $MT(\langle a_j \rangle _{j=1}^m, \langle x_n\rangle _{n})\subseteq A.$ upper M upper T left parenthesis left angle bracket a Subscript j Baseline right angle bracket Subscript j equals 1 Superscript m Baseline comma left angle bracket x Subscript n Baseline right angle bracket Subscript n Baseline right parenthesis subset of or equal to upper A period

Our main result is the following weaker version of Problem 1.3.

Theorem 1.6. Let $m\in \mathbb {N}$ m element of double struck upper N and a sequence $\langle a_j\rangle _{j=1}^m$ left angle bracket a Subscript j Baseline right angle bracket Subscript j equals 1 Superscript m in $\mathbb {N}$ double struck upper N be given. If $E \subset \mathbb {Z}^2$ upper E subset of double struck upper Z squared is a set of positive upper Banach density, then there exist infinitely many $k\in \mathbb {Z}$ k element of double struck upper Z and infinitely many sequences $\langle x_n\rangle _n$ left angle bracket x Subscript n Baseline right angle bracket Subscript n in $\mathbb {Z}$ double struck upper Z such that $ k\cdot MT(\langle a_j \rangle _{j=1}^m, \langle x_n\rangle _{n}) \subseteq \{ xy : (x,y) \in E - E \}. $ k dot upper M upper T left parenthesis left angle bracket a Subscript j Baseline right angle bracket Subscript j equals 1 Superscript m Baseline comma left angle bracket x Subscript n Baseline right angle bracket Subscript n Baseline right parenthesis subset of or equal to StartSet x y colon left parenthesis x comma y right parenthesis element of upper E minus upper E EndSet period

2 Preliminaries

2.1 Ergodic preliminaries

We study recurrence for countable amenable groups. The reason for restricting to countable amenable groups is that they provide a natural framework for Furstenberg’s correspondence principle.

Definition 2.1. A discrete group G is said to be amenable if there exists an invariant mean on the space $\mathcal {B}(G)$ script upper B left parenthesis upper G right parenthesis of real-valued bounded functions on G, that is, a positive linear functional $L : \mathcal {B}(G) \to \mathbb {R}$ upper L colon script upper B left parenthesis upper G right parenthesis right arrow double struck upper R satisfying:

  1. (1) $L(\mathbf {1}_G) = 1$ upper L left parenthesis bold 1 Subscript upper G Baseline right parenthesis equals 1 ;

  2. (2) $L(f_g) = L({}_g f) = L(f)$ upper L left parenthesis f Subscript g Baseline right parenthesis equals upper L left parenthesis Subscript g Baseline f right parenthesis equals upper L left parenthesis f right parenthesis for all $f \in \mathcal {B}(G)$ f element of script upper B left parenthesis upper G right parenthesis and all $g \in G$ g element of upper G , where $f_g(t) = f(tg)$ f Subscript g Baseline left parenthesis t right parenthesis equals f left parenthesis t g right parenthesis and ${}_g f(t) = f(gt)$ Subscript g Baseline f left parenthesis t right parenthesis equals f left parenthesis g t right parenthesis .

The existence of an invariant mean is only one of many equivalent characterisations of amenability. In what follows, we shall make frequent use of the Følner characterisation of amenability for discrete groups.

Theorem 2.2. A countable group G is amenable if and only if it admits a left Følner sequence, that is, a sequence of finite nonempty sets $(F_n)_{n \in \mathbb {N}} \subseteq G$ left parenthesis upper F Subscript n Baseline right parenthesis Subscript n element of double struck upper N Baseline subset of or equal to upper G such that for every $g \in G$ g element of upper G ,

$$ \begin{align*} \lim_{n \to \infty} \frac{|gF_n \triangle F_n|}{|F_n|} = 0. \end{align*} $$ limit Underscript n right arrow infinity Endscripts StartFraction StartAbsoluteValue g upper F Subscript n Baseline white up pointing triangle upper F Subscript n Baseline EndAbsoluteValue Over StartAbsoluteValue upper F Subscript n Baseline EndAbsoluteValue EndFraction equals 0 period

Definition 2.3. Let G be a countable amenable group and let $E $ upper E be a subset of G.

  1. (1) The positive Følner density, or positive upper density with respect to a left Følner sequence $(F_n)_{n \in \mathbb {N}}$ left parenthesis upper F Subscript n Baseline right parenthesis Subscript n element of double struck upper N , is defined by

    $$ \begin{align*} d_{F_n}(E) := \limsup_{n \to \infty} \frac{|E \cap F_n|}{|F_n|}> 0. \end{align*} $$ d Subscript upper F Sub Subscript n Baseline left parenthesis upper E right parenthesis colon equals limit sup Underscript n right arrow infinity Endscripts StartFraction StartAbsoluteValue upper E intersection upper F Subscript n Baseline EndAbsoluteValue Over StartAbsoluteValue upper F Subscript n Baseline EndAbsoluteValue EndFraction greater than 0 period

  2. (2) The upper Banach density of E is defined as $d^\star (E)=\sup _{(F_n)}d_{F_n}(E),$ d Superscript star Baseline left parenthesis upper E right parenthesis equals sup Underscript left parenthesis upper F Subscript n Baseline right parenthesis Endscripts d Subscript upper F Sub Subscript n Subscript Baseline left parenthesis upper E right parenthesis comma where the supremum is taken over all Følner sequences $(F_n)$ left parenthesis upper F Subscript n Baseline right parenthesis in G.

Note that central sets can be defined as members of minimal idempotent ultrafilters and central* sets are those sets that sit in every minimal idempotent ultrafilter. We will recall these notions in the next section. The following theorem, which follows from [Reference Bergelson and McCutcheon1, Theorem 1.4], shows that the set of return times is large.

Theorem 2.4. Let G be a countable amenable group, let $\{T_g\}_{g \in G}$ left brace upper T Subscript g Baseline right brace Subscript g element of upper G and $\{S_g\}_{g \in G}$ left brace upper S Subscript g Baseline right brace Subscript g element of upper G be commuting measure preserving actions of G on a probability space $(X,\mathcal {B},\mu )$ left parenthesis upper X comma script upper B comma mu right parenthesis , and let $A \in \mathcal {B}$ upper A element of script upper B satisfy $\mu (A)> 0$ mu left parenthesis upper A right parenthesis greater than 0 . Then, the set

$$ \begin{align*} \{ g \in G : \mu(A \cap T_g^{-1}A \cap (T_g S_g)^{-1}A)> 0 \} \end{align*} $$ StartSet g element of upper G colon mu left parenthesis upper A intersection upper T Subscript g Superscript negative 1 Baseline upper A intersection left parenthesis upper T Subscript g Baseline upper S Subscript g Baseline right parenthesis Superscript negative 1 Baseline upper A right parenthesis greater than 0 EndSet

is central* in G.

From the above theorem, it is also clear that $\{ g \in G : \mu (A \cap S_g^{-1}A \cap (T_g S_g)^{-1}A)> 0 \}$ StartSet g element of upper G colon mu left parenthesis upper A intersection upper S Subscript g Superscript negative 1 Baseline upper A intersection left parenthesis upper T Subscript g Baseline upper S Subscript g Baseline right parenthesis Superscript negative 1 Baseline upper A right parenthesis greater than 0 EndSet is central* in G. The Furstenberg correspondence principle over amenable groups and Theorem 2.4 have the following corollary.

Corollary 2.5. Let G be a countable left amenable group and let $E\subseteq G\times G$ upper E subset of or equal to upper G times upper G be such that $d^\star (E)>0.$ d Superscript star Baseline left parenthesis upper E right parenthesis greater than 0 period Then,

$$ \begin{align*} \{x\in G: d(E \cap (x,e)^{-1}E \cap (x,x)^{-1}E)> 0\} \end{align*} $$ StartSet x element of upper G colon d left parenthesis upper E intersection left parenthesis x comma e right parenthesis Superscript negative 1 Baseline upper E intersection left parenthesis x comma x right parenthesis Superscript negative 1 Baseline upper E right parenthesis greater than 0 EndSet

is a central* set. And similarly,

$$ \begin{align*} \{x\in G: d(E \cap (e,x)^{-1}E \cap (x,x)^{-1}E)> 0\} \end{align*} $$ StartSet x element of upper G colon d left parenthesis upper E intersection left parenthesis e comma x right parenthesis Superscript negative 1 Baseline upper E intersection left parenthesis x comma x right parenthesis Superscript negative 1 Baseline upper E right parenthesis greater than 0 EndSet

is a central* set.

Next, we define the notions of density recurrent sets, which will be useful in our study.

Definition 2.6. Let G be a countable left amenable group.

  1. (1) Let $B \subseteq G$ upper B subset of or equal to upper G . Then, B is left density 2-recurrent if and only if whenever $E \subseteq G \times G$ upper E subset of or equal to upper G times upper G satisfies $d(E)> 0$ d left parenthesis upper E right parenthesis greater than 0 , there exists $x \in B$ x element of upper B such that

    $$ \begin{align*} d(E \cap (x,e)^{-1}E \cap (x,x)^{-1}E)> 0. \end{align*} $$ d left parenthesis upper E intersection left parenthesis x comma e right parenthesis Superscript negative 1 Baseline upper E intersection left parenthesis x comma x right parenthesis Superscript negative 1 Baseline upper E right parenthesis greater than 0 period

  2. (2) Let $C \subseteq G$ upper C subset of or equal to upper G . Then, C is right density 2-recurrent if and only if whenever $E \subseteq G \times G$ upper E subset of or equal to upper G times upper G satisfies $d(E)> 0$ d left parenthesis upper E right parenthesis greater than 0 , there exists $x \in C$ x element of upper C such that

    $$ \begin{align*} d(E \cap (e,x)^{-1}E \cap (x,x)^{-1}E)> 0. \end{align*} $$ d left parenthesis upper E intersection left parenthesis e comma x right parenthesis Superscript negative 1 Baseline upper E intersection left parenthesis x comma x right parenthesis Superscript negative 1 Baseline upper E right parenthesis greater than 0 period

  3. (3) Let $D \subseteq G$ upper D subset of or equal to upper G . Then, D is both-sided density 2-recurrent if and only if whenever $E \subseteq G \times G$ upper E subset of or equal to upper G times upper G satisfies $d(E)> 0$ d left parenthesis upper E right parenthesis greater than 0 , there exists $x \in D$ x element of upper D such that:

    1. (a) $d(E \cap (e,x)^{-1}E \cap (x,x)^{-1}E)> 0$ d left parenthesis upper E intersection left parenthesis e comma x right parenthesis Superscript negative 1 Baseline upper E intersection left parenthesis x comma x right parenthesis Superscript negative 1 Baseline upper E right parenthesis greater than 0 ; and

    2. (b) $d(E \cap (x,e)^{-1}E \cap (x,x)^{-1}E)> 0.$ d left parenthesis upper E intersection left parenthesis x comma e right parenthesis Superscript negative 1 Baseline upper E intersection left parenthesis x comma x right parenthesis Superscript negative 1 Baseline upper E right parenthesis greater than 0 period

2.2 A brief review of topological algebra

Here, we recall some basic facts about ultrafilters. For details, we refer to the book [Reference Hindman and Strauss7].

Let $(S,\cdot )$ left parenthesis upper S comma dot right parenthesis be any discrete semigroup. Let $\beta S$ beta upper S be the collection of all ultrafilters on S. For every $A\subseteq S,$ upper A subset of or equal to upper S comma define $\overline {A}=\{p\in \beta S: A\in p\}.$ upper A overbar equals StartSet p element of beta upper S colon upper A element of p EndSet period One can check that the collection ${\{\overline {A}: A\subseteq S\}}$ StartSet upper A overbar colon upper A subset of or equal to upper S EndSet forms a basis for a topology. This basis generates a topology over $\beta S.$ beta upper S period We can extend the operation ‘ $\cdot $ dot ’ of S over $\beta S$ beta upper S as follows: for any $p,q\in \beta S,\, A\in p\cdot q$ p comma q element of beta upper S comma upper A element of p dot q if and only if $\{x:x^{-1}A\in q\}\in p,$ StartSet x colon x Superscript negative 1 Baseline upper A element of q EndSet element of p comma where $x^{-1}A=\{y\in S: xy\in A\}.$ x Superscript negative 1 Baseline upper A equals StartSet y element of upper S colon x y element of upper A EndSet period With this operation ‘ $\cdot $ dot ’, $(\beta S,\cdot )$ left parenthesis beta upper S comma dot right parenthesis becomes a compact Hausdorff right topological semigroup. One can show that $\beta S$ beta upper S is nothing but the Stone–Čech compactification of $S.$ upper S period Hence, Ellis’s theorem guarantees that there exist idempotents in $(\beta S,\cdot )$ left parenthesis beta upper S comma dot right parenthesis . The set of all idempotents in $(\beta S,\cdot )$ left parenthesis beta upper S comma dot right parenthesis is denoted by $E((\beta S,\cdot )).$ upper E left parenthesis left parenthesis beta upper S comma dot right parenthesis right parenthesis period It can be shown that every member of the idempotents of $(\beta S,\cdot )$ left parenthesis beta upper S comma dot right parenthesis contains an IP set, which means every idempotent witnesses Hindman’s theorem. Using Zorn’s lemma, one can show that $(\beta S,\cdot )$ left parenthesis beta upper S comma dot right parenthesis contains minimal left ideals (minimal with respect to the inclusion). A well-known fact is that the union of such minimal left ideals is a minimal two-sided ideal, denoted by $K(\beta S,\cdot ).$ upper K left parenthesis beta upper S comma dot right parenthesis period If S is a group, then for any ultrafilter p, define $p^{-1}=\{A^{-1}:A\in p\}.$ p Superscript negative 1 Baseline equals StartSet upper A Superscript negative 1 Baseline colon upper A element of p EndSet period It can be verified that $p^{-1}$ p Superscript negative 1 is also an ultrafilter. Here, we recall a few well-known classes of sets that are relevant to our work.

Definition 2.7.

  1. (1) The set A is a central set if it belongs to a minimal idempotent in $\beta S$ beta upper S .

  2. (2) The set A is a central* set if it belongs to all minimal idempotents in $\beta S$ beta upper S .

The tensor product of two ultrafilters is defined as follows.

Definition 2.8. Let $(S,\cdot )$ left parenthesis upper S comma dot right parenthesis and $(T,\cdot )$ left parenthesis upper T comma dot right parenthesis be two discrete semigroups. Let $p\in (\beta S,\cdot )$ p element of left parenthesis beta upper S comma dot right parenthesis and $q\in (\beta T,\cdot )$ q element of left parenthesis beta upper T comma dot right parenthesis be two ultrafilters. Then, the tensor product of p and q is defined by

$$ \begin{align*}p\otimes q=\{A\subseteq S\times T: \{x\in S: \{y:(x,y)\in A\}\in q\}\in p\},\end{align*} $$ p circled times q equals StartSet upper A subset of or equal to upper S times upper T colon StartSet x element of upper S colon StartSet y colon left parenthesis x comma y right parenthesis element of upper A EndSet element of q EndSet element of p EndSet comma

where $x\in S$ x element of upper S , $y\in T.$ y element of upper T period

3 Our results

For any countable amenable group $(G,\cdot ),$ left parenthesis upper G comma dot right parenthesis comma define the following notions.

Definition 3.1. Define:

  1. (1) $\mathcal {LDR}_2(G)=\{ p \in \beta G : (\text {for all } B \in p)\; B \text { is left density } 2\text {-recurrent} \};$ script upper L upper D upper R 2 left parenthesis upper G right parenthesis equals StartSet p element of beta upper G colon left parenthesis for all upper B element of p right parenthesis upper B is left density 2 minus recurrent EndSet semicolon

  2. (2) $\mathcal {RDR}_2(G)=\{ p \in \beta G : (\text {for all } B \in p)\; B \text { is right density } 2\text {-recurrent} \};$ script upper R upper D upper R 2 left parenthesis upper G right parenthesis equals StartSet p element of beta upper G colon left parenthesis for all upper B element of p right parenthesis upper B is right density 2 minus recurrent EndSet semicolon

  3. (3) $\mathcal {DR}_2(G)=\{ p \in \beta G : (\text {for all } B \in p)\; B \text { is both-sided density } 2\text {-recurrent} \}.$ script upper D upper R 2 left parenthesis upper G right parenthesis equals StartSet p element of beta upper G colon left parenthesis for all upper B element of p right parenthesis upper B is both minus sided density 2 minus recurrent EndSet period

To obtain Theorem 1.6, we will study the properties of the set $AA^{-1}$ upper A upper A Superscript negative 1 , where ${A\subseteq G\times G}$ upper A subset of or equal to upper G times upper G has positive upper Banach density. First, we prove some lemmas.

Lemma 3.2. Let G be a countable amenable group. Then, both $ \mathcal {LDR}_2(G)$ script upper L upper D upper R 2 left parenthesis upper G right parenthesis and $\mathcal {RDR}_2(G)$ script upper R upper D upper R 2 left parenthesis upper G right parenthesis are subsemigroups of $\beta G$ beta upper G containing the minimal idempotents of $\beta G$ beta upper G .

Proof. We prove the result $\mathcal {LDR}_2(G).$ script upper L upper D upper R 2 left parenthesis upper G right parenthesis period The proof for $\mathcal {RDR}_2(G)$ script upper R upper D upper R 2 left parenthesis upper G right parenthesis is similar.

By Corollary 2.5, whenever $E \subseteq G \times G$ upper E subset of or equal to upper G times upper G satisfies $d^\star (E)> 0$ d Superscript star Baseline left parenthesis upper E right parenthesis greater than 0 , the set

$$ \begin{align*} \{ x \in G : d^\star(E \cap (x,e)^{-1}E \cap (x,x)^{-1}E)> 0 \} \end{align*} $$ StartSet x element of upper G colon d Superscript star Baseline left parenthesis upper E intersection left parenthesis x comma e right parenthesis Superscript negative 1 Baseline upper E intersection left parenthesis x comma x right parenthesis Superscript negative 1 Baseline upper E right parenthesis greater than 0 EndSet

is central*. Let p be a minimal idempotent in $\beta G$ beta upper G and let $C \in p$ upper C element of p . Then, for every ${E \subseteq G \times G}$ upper E subset of or equal to upper G times upper G with $d^\star (E)> 0$ d Superscript star Baseline left parenthesis upper E right parenthesis greater than 0 ,

$$ \begin{align*} C \cap \{ x \in G : d^\star(E \cap (x,e)^{-1}E \cap (x,x)^{-1}E)> 0 \} \neq \varnothing. \end{align*} $$ upper C intersection StartSet x element of upper G colon d Superscript star Baseline left parenthesis upper E intersection left parenthesis x comma e right parenthesis Superscript negative 1 Baseline upper E intersection left parenthesis x comma x right parenthesis Superscript negative 1 Baseline upper E right parenthesis greater than 0 EndSet not equals empty set period

Hence, there exists $g \in C$ g element of upper C such that $ d^\star (E \cap (g,e)^{-1}E \cap (g,g)^{-1}E)> 0. $ d Superscript star Baseline left parenthesis upper E intersection left parenthesis g comma e right parenthesis Superscript negative 1 Baseline upper E intersection left parenthesis g comma g right parenthesis Superscript negative 1 Baseline upper E right parenthesis greater than 0 period Thus, C is density $2$ 2 -recurrent, which implies $K(\beta G) \subseteq \mathcal {DR}_2(G)$ upper K left parenthesis beta upper G right parenthesis subset of or equal to script upper D upper R 2 left parenthesis upper G right parenthesis . In particular, $\mathcal {DR}_2(G)$ script upper D upper R 2 left parenthesis upper G right parenthesis is nonempty.

Now, let $p,q \in \mathcal {DR}_2(G)$ p comma q element of script upper D upper R 2 left parenthesis upper G right parenthesis . Let $B \in pq$ upper B element of p q and $A \subseteq G \times G$ upper A subset of or equal to upper G times upper G with $d^\star (A)> 0$ d Superscript star Baseline left parenthesis upper A right parenthesis greater than 0 . Define

$$ \begin{align*} C = \{ x \in G : x^{-1}B \in q \}. \end{align*} $$ upper C equals StartSet x element of upper G colon x Superscript negative 1 Baseline upper B element of q EndSet period

Then, $C \in p$ upper C element of p and, hence, C is density $2$ 2 -recurrent. Choose $x \in C$ x element of upper C such that

$$ \begin{align*} d^\star(A \cap (x,e)^{-1}A \cap (x,x)^{-1}A)> 0 \end{align*} $$ d Superscript star Baseline left parenthesis upper A intersection left parenthesis x comma e right parenthesis Superscript negative 1 Baseline upper A intersection left parenthesis x comma x right parenthesis Superscript negative 1 Baseline upper A right parenthesis greater than 0

and set $ D = A \cap (x,e)^{-1}A \cap (x,x)^{-1}A. $ upper D equals upper A intersection left parenthesis x comma e right parenthesis Superscript negative 1 Baseline upper A intersection left parenthesis x comma x right parenthesis Superscript negative 1 Baseline upper A period Since $x^{-1}B \in q$ x Superscript negative 1 Baseline upper B element of q , there exists $y \in x^{-1}B$ y element of x Superscript negative 1 Baseline upper B such that

$$ \begin{align*} d^\star(D \cap (y,e)^{-1}D \cap (y,y)^{-1}D)> 0. \end{align*} $$ d Superscript star Baseline left parenthesis upper D intersection left parenthesis y comma e right parenthesis Superscript negative 1 Baseline upper D intersection left parenthesis y comma y right parenthesis Superscript negative 1 Baseline upper D right parenthesis greater than 0 period

As $xy \in B$ x y element of upper B , it suffices to show that

$$ \begin{align*} D \cap (y,e)^{-1}D \cap (y,y)^{-1}D \subseteq A \cap (xy,e)^{-1}A \cap (xy,xy)^{-1}A. \end{align*} $$ upper D intersection left parenthesis y comma e right parenthesis Superscript negative 1 Baseline upper D intersection left parenthesis y comma y right parenthesis Superscript negative 1 Baseline upper D subset of or equal to upper A intersection left parenthesis x y comma e right parenthesis Superscript negative 1 Baseline upper A intersection left parenthesis x y comma x y right parenthesis Superscript negative 1 Baseline upper A period

Let $(u,v)$ left parenthesis u comma v right parenthesis belong to the left-hand side. Then, $(u,v) \in D \subseteq A$ left parenthesis u comma v right parenthesis element of upper D subset of or equal to upper A and

$$ \begin{align*} (u,v) \in (y,e)^{-1}D \subseteq (y,e)^{-1}(x,e)^{-1}A, \end{align*} $$ left parenthesis u comma v right parenthesis element of left parenthesis y comma e right parenthesis Superscript negative 1 Baseline upper D subset of or equal to left parenthesis y comma e right parenthesis Superscript negative 1 Baseline left parenthesis x comma e right parenthesis Superscript negative 1 Baseline upper A comma

so $(xyu,v) \in A$ left parenthesis x y u comma v right parenthesis element of upper A , which implies $(u,v) \in (xy,e)^{-1}A$ left parenthesis u comma v right parenthesis element of left parenthesis x y comma e right parenthesis Superscript negative 1 Baseline upper A . Similarly,

$$ \begin{align*} (u,v) \in (y,y)^{-1}D \subseteq (y,y)^{-1}(x,x)^{-1}A, \end{align*} $$ left parenthesis u comma v right parenthesis element of left parenthesis y comma y right parenthesis Superscript negative 1 Baseline upper D subset of or equal to left parenthesis y comma y right parenthesis Superscript negative 1 Baseline left parenthesis x comma x right parenthesis Superscript negative 1 Baseline upper A comma

so $(xyu,xyv) \in A$ left parenthesis x y u comma x y v right parenthesis element of upper A and, hence, $(u,v) \in (xy,xy)^{-1}A$ left parenthesis u comma v right parenthesis element of left parenthesis x y comma x y right parenthesis Superscript negative 1 Baseline upper A . Therefore,

$$ \begin{align*} (u,v) \in A \cap (xy,e)^{-1}A \cap (xy,xy)^{-1}A. \\[-33pt] \end{align*} $$ left parenthesis u comma v right parenthesis element of upper A intersection left parenthesis x y comma e right parenthesis Superscript negative 1 Baseline upper A intersection left parenthesis x y comma x y right parenthesis Superscript negative 1 Baseline upper A period

Note that from Corollary 2.5, it is clear that $E(K(\beta G,\cdot ))\subseteq \mathcal {DR}_2(G).$ upper E left parenthesis upper K left parenthesis beta upper G comma dot right parenthesis right parenthesis subset of or equal to script upper D upper R 2 left parenthesis upper G right parenthesis period

As both $\mathcal {LDR}_2(G)$ script upper L upper D upper R 2 left parenthesis upper G right parenthesis and $ \mathcal {RDR}_2(G)$ script upper R upper D upper R 2 left parenthesis upper G right parenthesis contain $ \mathcal {DR}_2(G),$ script upper D upper R 2 left parenthesis upper G right parenthesis comma from Lemma 3.2, it is clear that if $p\in \mathcal {DR}_2(G)$ p element of script upper D upper R 2 left parenthesis upper G right parenthesis and $q\in \mathcal {LDR}_2(G),$ q element of script upper L upper D upper R 2 left parenthesis upper G right parenthesis comma then $pq\in \mathcal {LDR}_2(G).$ p q element of script upper L upper D upper R 2 left parenthesis upper G right parenthesis period The same conclusion holds for $ \mathcal {RDR}_2(G).$ script upper R upper D upper R 2 left parenthesis upper G right parenthesis period

Let

$$ \begin{align*}T=\bigcap \lbrace \overline{EE^{-1}}: A\subseteq G\times G \text{ and } d^\star(E)>0 \rbrace.\end{align*} $$ upper T equals intersection StartSet ModifyingAbove upper E upper E Superscript negative 1 Baseline With quotation dash colon upper A subset of or equal to upper G times upper G and d Superscript star Baseline left parenthesis upper E right parenthesis greater than 0 EndSet period

The following lemma shows that the tensor product of two elements from $\mathcal {DR}_2(G)$ script upper D upper R 2 left parenthesis upper G right parenthesis is a member of $T.$ upper T period

Lemma 3.3. If $p,q\in \mathcal {DR}_2(G),$ p comma q element of script upper D upper R 2 left parenthesis upper G right parenthesis comma then $ p\otimes q\in T.$ p circled times q element of upper T period

Proof. For a contradiction, assume that $p\otimes q\notin T.$ p circled times q not an element of upper T period Then, there exists a set A with $d(A)>0$ d left parenthesis upper A right parenthesis greater than 0 such that $A^{-1}A\notin p\otimes q.$ upper A Superscript negative 1 Baseline upper A not an element of p circled times q period Hence, $B=(A^{-1}A)^c\in p\otimes q.$ upper B equals left parenthesis upper A Superscript negative 1 Baseline upper A right parenthesis Superscript c Baseline element of p circled times q period However, ${B\in p\otimes q}$ upper B element of p circled times q implies $C=\{x:\{y:(x,y)\in B\}\in q\}\in p.$ upper C equals StartSet x colon StartSet y colon left parenthesis x comma y right parenthesis element of upper B EndSet element of q EndSet element of p period Let $D=A\cap (x,e)^{-1}A\cap (x,x)^{-1}A$ upper D equals upper A intersection left parenthesis x comma e right parenthesis Superscript negative 1 Baseline upper A intersection left parenthesis x comma x right parenthesis Superscript negative 1 Baseline upper A . Choose $x\in C$ x element of upper C with $d(D)>0$ d left parenthesis upper D right parenthesis greater than 0 .

Now, there exists $y\in B_x=\{(y:(x,y)\in B\}$ y element of upper B Subscript x Baseline equals left brace left parenthesis y colon left parenthesis x comma y right parenthesis element of upper B right brace such that ${d(D\cap (e,y)^{-1}D\cap (y,y)^{-1}D)>0.}$ d left parenthesis upper D intersection left parenthesis e comma y right parenthesis Superscript negative 1 Baseline upper D intersection left parenthesis y comma y right parenthesis Superscript negative 1 Baseline upper D right parenthesis greater than 0 period However, this implies $d(A\cap (x,y)^{-1}A)>0$ d left parenthesis upper A intersection left parenthesis x comma y right parenthesis Superscript negative 1 Baseline upper A right parenthesis greater than 0 and so $(x,y)\in A^{-1}A\cap B,$ left parenthesis x comma y right parenthesis element of upper A Superscript negative 1 Baseline upper A intersection upper B comma which is a contradiction.

The above proof suggests something more, leading to the following lemma.

Lemma 3.4. If $p\in \mathcal {LDR}_2(G)$ p element of script upper L upper D upper R 2 left parenthesis upper G right parenthesis and $ q\in \mathcal {RDR}_2(G),$ q element of script upper R upper D upper R 2 left parenthesis upper G right parenthesis comma then $p\otimes q\in T.$ p circled times q element of upper T period

Thus, if $E\subseteq \mathbb {Z}\times \mathbb {Z}$ upper E subset of or equal to double struck upper Z times double struck upper Z is a subset with positive upper Banach density, then for any ${p\in \mathcal {LDR}_2(G)}$ p element of script upper L upper D upper R 2 left parenthesis upper G right parenthesis and $ q\in \mathcal {RDR}_2(G),$ q element of script upper R upper D upper R 2 left parenthesis upper G right parenthesis comma we have $(E-E)\in p\otimes q.$ left parenthesis upper E minus upper E right parenthesis element of p circled times q period

To prove Theorem 1.6, we need the following lemma. Again, the reference contains the proof over the set of positive integers, but the same proof holds over the set of all integers.

Lemma 3.5 [Reference Hindman and Strauss7, Lemma 5.19.2].

If $p{\kern-1pt}\in{\kern-1pt} E(K(\beta \mathbb {Z},+))$ p element of upper E left parenthesis upper K left parenthesis beta double struck upper Z comma plus right parenthesis right parenthesis and $n{\kern-1pt}\in{\kern-1pt} \mathbb {N},$ n element of double struck upper N comma then ${n\cdot p{\kern-1pt}\in{\kern-1pt} E(K(\beta \mathbb {Z},+)).}$ n dot p element of upper E left parenthesis upper K left parenthesis beta double struck upper Z comma plus right parenthesis right parenthesis period

Now, we prove our theorem.

Proof of Theorem 1.6.

Define $f:\mathbb {Z}^2\to \mathbb {Z}$ f colon double struck upper Z squared right arrow double struck upper Z by $f(x,y)=xy$ f left parenthesis x comma y right parenthesis equals x y and let ${\beta f:\beta \mathbb {Z}^2\to \beta \mathbb {Z}}$ beta f colon beta double struck upper Z squared right arrow beta double struck upper Z be its continuous extension. First, we claim that $\beta f(p\otimes q)=p\cdot q.$ beta f left parenthesis p circled times q right parenthesis equals p dot q period As both are ultrafilters, it will be sufficient to show that $\beta f(p\otimes q) \subseteq p\cdot q.$ beta f left parenthesis p circled times q right parenthesis subset of or equal to p dot q period Suppose $B\in \beta f(p\otimes q).$ upper B element of beta f left parenthesis p circled times q right parenthesis period Then, $f^{-1}(B)\in p\otimes q$ f Superscript negative 1 Baseline left parenthesis upper B right parenthesis element of p circled times q and $\{a:\{b: (a,b)\in f^{-1}(B)\}\in q\}\in p.$ StartSet a colon StartSet b colon left parenthesis a comma b right parenthesis element of f Superscript negative 1 Baseline left parenthesis upper B right parenthesis EndSet element of q EndSet element of p period However, $(a,b)\in f^{-1}(B)$ left parenthesis a comma b right parenthesis element of f Superscript negative 1 Baseline left parenthesis upper B right parenthesis implies $a\cdot b\in B$ a dot b element of upper B , which implies $B\in p\cdot q.$ upper B element of p dot q period This proves the claim.

Now, choose any $p,q\in E(K(\beta \mathbb {Z},+))$ p comma q element of upper E left parenthesis upper K left parenthesis beta double struck upper Z comma plus right parenthesis right parenthesis . From Lemmas 3.5 and 3.2, $a_1q+\cdots +a_mq\in \mathcal {RDR}_2(\mathbb {Z}).$ a 1 q plus midline horizontal ellipsis plus a Subscript m Baseline q element of script upper R upper D upper R 2 left parenthesis double struck upper Z right parenthesis period However, from Lemma 3.4 and our claim, it follows that

$$ \begin{align*}f(E-E)\in \beta f(p\otimes (a_1q+\cdots +a_mq))=p\cdot (a_1q+\cdots +a_mq).\end{align*} $$ f left parenthesis upper E minus upper E right parenthesis element of beta f left parenthesis p circled times left parenthesis a 1 q plus midline horizontal ellipsis plus a Subscript m Baseline q right parenthesis right parenthesis equals p dot left parenthesis a 1 q plus midline horizontal ellipsis plus a Subscript m Baseline q right parenthesis period

Hence,

$$ \begin{align*}\{ xy : (x,y) \in E - E \}\in p\cdot (a_1q+\cdots +a_mq).\end{align*} $$ StartSet x y colon left parenthesis x comma y right parenthesis element of upper E minus upper E EndSet element of p dot left parenthesis a 1 q plus midline horizontal ellipsis plus a Subscript m Baseline q right parenthesis period

Now, from Theorem 1.5, it is a routine exercise to check that there are infinitely many sequences $\langle x_n\rangle _n$ left angle bracket x Subscript n Baseline right angle bracket Subscript n in $\mathbb {Z}$ double struck upper Z such that

$$ \begin{align*} k\cdot MT(\langle a_j \rangle_{j=1}^m, \langle x_n\rangle_{n}) \subseteq \{ xy : (x,y) \in E - E \}. \end{align*} $$ k dot upper M upper T left parenthesis left angle bracket a Subscript j Baseline right angle bracket Subscript j equals 1 Superscript m Baseline comma left angle bracket x Subscript n Baseline right angle bracket Subscript n Baseline right parenthesis subset of or equal to StartSet x y colon left parenthesis x comma y right parenthesis element of upper E minus upper E EndSet period

Footnotes

The author is supported by NBHM postdoctoral fellowship with reference nos. 0204/27/(27)/2023/R and D-II/11927.

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