Popular Differences for Corners in Abelian Groups

TL;DR

This paper extends the 'popular differences' theorem for corners from finite fields to all compact abelian groups and the integer lattice, showing large subsets contain many structured configurations with fixed differences.

Contribution

It generalizes Mandache's finite field result to broader classes of groups, including all compact abelian groups and the integer lattice, revealing new structural insights.

Findings

Generalization of the popular differences theorem to all compact abelian groups

Extension of the result to corners in ^2 and ^2

Large subsets contain a significant fraction of corners with fixed differences

Abstract

For a compact abelian group $G$, a corner in $G \times G$ is a triple of points $(x,y)$, $(x,y+d)$, $(x+d,y)$. The classical corners theorem of Ajtai and Szemer\'edi implies that for every $\alpha > 0$, there is some $\delta > 0$ such that every subset $A \subset G \times G$ of density $\alpha$ contains a $\delta$ fraction of all corners in $G \times G$, as $x,y,d$ range over $G$. Recently, Mandache proved a "popular differences" version of this result in the finite field case $G = \mathbb F_p^n$, showing that for any subset $A \subset G \times G$ of density $\alpha$, one can fix $d \neq 0$ such that $A$ contains a large fraction, now known to be approximately $\alpha^4$, of all corners with difference $d$, as $x,y$ vary over $G$. We generalize Mandache's result to all compact abelian groups $G$, as well as the case of corners in $\mathbb Z^2$.