Popular differences for right isosceles triangles

TL;DR

This paper proves the existence of certain difference patterns in large subsets of integer grids, extending results to compact abelian groups and finite fields, and answers a previously posed question.

Contribution

It introduces a new approach to find difference patterns in large subsets, generalizing prior methods for three-term arithmetic progressions.

Findings

Existence of difference triples in large subsets of integer grids.

Extension of results to compact abelian groups.

Comments on the rarity of such configurations in finite fields.

Abstract

For a subset $A$ of $\{1,2,\ldots,N\}^2$ of size $\alpha N^2$ we show existence of $(m,n)\neq(0,0)$ such that the set $A$ contains at least $(\alpha^3 - o(1))N^2$ triples of points of the form $(a,b)$, $(a+m,b+n)$, $(a-n,b+m)$. This answers a question by Ackelsberg, Bergelson, and Best from arXiv:2101.02811. The same approach also establishes the corresponding result for compact abelian groups. Furthermore, for a finite field $\mathbb{F}_q$ we comment on exponential smallness of subsets of $(\mathbb{F}_q^n)^2$ that avoid the aforementioned configuration. The proofs are minor modifications of the existing proofs regarding three-term arithmetic progressions.