Popular differences for matrix patterns

Abstract: We study matrix patterns including those of the form $x,x+M_1d,x+M_2d,x+(M_1+M_2)d$ in abelian groups $G^k$ for integer matrices $M_1,M_2$. If $A\subseteq G^k$ has density $\alpha$, one might expect based on recent conjectures of Ackelsberg, Bergelson, and Best that there is $d\neq 0$ such that \[\#\{x \in G^k: x, x+M_1d, x+M_2d, x+(M_1+M_2)d \in A\} \ge (\alpha^4-o(1))|G|^k\] as long as $M_1,M_2,M_1\pm M_2$ define automorphisms of $G^k$. We show this conjecture holds in $G = \mathbb{F}_p^n$ for odd $p$ given an additional spectral condition, but is false without this condition. Explicitly, we show the rotated squares pattern is false over $\mathbb{F}_5^n$. This is in surprising contrast to the theory of popular differences of one-dimensional patterns.

combinatoricsprobability

Audience: researchers in the topic

Extremal and probabilistic combinatorics webinar

Series comments: We've added a password: concatenate the 6 first prime numbers (hence obtaining an 8-digit password).