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On difference sets of dense subsets of $\mathbb{Z}^2$

Sayan Goswami

TL;DR

This paper investigates the difference set $E-E$ for dense subsets $E$ of $\mathbb{Z}^2$, addressing Fish's question about embedding a full arithmetic progression via products. It employs ultrafilter techniques on the Stone-Čech compactification and leverages IP-sets, central* sets, and tensor products to translate density information into multiplicative structure in $E-E$. The main contributions show that there exist infinitely many scalars $k$ and infinite IP-sets whose finite sums, scaled by $k$, lie in $\{xy:(x,y)\in E-E\}$, along with a related pattern of differences. These results advance infinitary sum-product phenomena in two dimensions and demonstrate the power of ultrafilter methods for detecting structured patterns in dense combinatorial objects.

Abstract

In this article, we study the structure of the difference set $E - E$ for subsets $E \subseteq \mathbb{Z}^2$ of positive upper Banach density. In [Proc. Amer. Math. Soc. 146 (2018), 3449-3453, Problem 2] Fish asked 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 of this conjecture. Specifically, we prove that there exist infinitely many integers $k \in \mathbb{Z}$ and a sequence $\langle x_n \rangle_{n \in \mathbb{N}}$ in $\mathbb{Z}$ such that \[ k \cdot FS(\langle x_n \rangle_{n \in \mathbb{N}}) \subseteq \{\, xy : (x,y) \in E - E \,\}, \] where $FS(\langle x_n \rangle)$ denotes the finite sums set generated by the sequence.

On difference sets of dense subsets of $\mathbb{Z}^2$

TL;DR

This paper investigates the difference set for dense subsets of , addressing Fish's question about embedding a full arithmetic progression via products. It employs ultrafilter techniques on the Stone-Čech compactification and leverages IP-sets, central* sets, and tensor products to translate density information into multiplicative structure in . The main contributions show that there exist infinitely many scalars and infinite IP-sets whose finite sums, scaled by , lie in , along with a related pattern of differences. These results advance infinitary sum-product phenomena in two dimensions and demonstrate the power of ultrafilter methods for detecting structured patterns in dense combinatorial objects.

Abstract

In this article, we study the structure of the difference set for subsets of positive upper Banach density. In [Proc. Amer. Math. Soc. 146 (2018), 3449-3453, Problem 2] Fish asked whether, for every such set , there exists a nonzero integer such that Although this question remains open, we establish a relatively weaker form of this conjecture. Specifically, we prove that there exist infinitely many integers and a sequence in such that where denotes the finite sums set generated by the sequence.
Paper Structure (5 sections, 9 theorems, 29 equations)

This paper contains 5 sections, 9 theorems, 29 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Ergodic preliminaries
  4. A brief review of Topological Algebra
  5. Our Results

Key Result

Theorem 1.4

If $E \subset \mathbb{Z}^2$ is a set of positive upper Banach density, then there exist infinitely many $k\in \mathbb{Z}$ and infinitely many sequences $\langle x_n\rangle_n$ in $\mathbb{Z}$ such that

Theorems & Definitions (22)

  • Conjecture 1.1: Erdős--Szemerédi erdos
  • Definition 1.2: Upper Banach density in $\mathbb{Z}^2$
  • Theorem 1.4
  • Theorem 1.5
  • Definition 2.1
  • Theorem 2.2
  • Definition 2.3
  • Theorem 2.4
  • Corollary 2.5
  • Definition 2.6
  • ...and 12 more