Popular differences for matrix patterns

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 \[\#\{x \in G^k: x, x+M_1d, x+M_2d, x+(M_1+M_2)d \in A\} \ge (α^4-o(1))N^k.\] 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 $M_1{M}_2^{-1}$ (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.

24 pages

References in corpus (2)

• A Szemeredi-type regularity lemma in abelian groups, with applications
• Montreal Lecture Notes on Quadratic Fourier Analysis

Cited by in corpus (4)

• Khintchine-type recurrence for 3-point configurations
• Khintchine-type double recurrence in abelian groups
• Bounds in a popular multidimensional nonlinear Roth theorem
• Multiple recurrence and popular differences for polynomial patterns in rings of integers