Popular differences for matrix patterns

TL;DR

This paper investigates a combinatorial conjecture about popular differences in matrix patterns within finite abelian groups, showing it generally fails but holds under specific spectral conditions in vector spaces over finite fields.

Contribution

It demonstrates the conjecture's failure in general and establishes its validity for certain matrix patterns over finite fields with spectral restrictions.

Findings

The conjecture is false in general.

The conjecture holds for G=F_p^n with spectral conditions.

Counterexamples exist for rotated squares pattern in F_5^n.

Abstract

The following combinatorial conjecture arises naturally from recent ergodic-theoretic work of Ackelsberg, Bergelson, and Best. Let $M_1$, $M_2$ be $k\times k$ integer matrices, $G$ be a finite abelian group of order $N$, and $A\subseteq G^k$ with $|A|\ge\alpha N^k$. If $M_1$, $M_2$, $M_1-M_2$, and $M_1+M_2$ are automorphisms of $G^k$, is it true that there exists a popular difference $d \in G^k\setminus\{0\}$ such that \[\#\{x \in G^k: x, x+M_1d, x+M_2d, x+(M_1+M_2)d \in A\} \ge (\alpha^4-o(1))N^k.\] We show that this conjecture is false in general, but holds for $G = \mathbb{F}_p^n$ with $p$ an odd prime given the additional spectral condition that no pair of eigenvalues of $M_1M_2^{-1}$ (over $\overline{\mathbb{F}}_p$) are negatives of each other. In particular, the "rotated squares" pattern does not satisfy this eigenvalue condition, and we give a construction of a set of positive density in $(\mathbb{F}_5^n)^2$ for which that pattern has no nonzero popular difference. This is in surprising contrast to three-point patterns, which we handle over all compact abelian groups and which do not require an additional spectral condition.