Common and Sidorenko Linear Equations

Abstract

A linear equation with coefficients in ${\mathbb{F}}_q$ is common if the number of monochromatic solutions in any two-coloring of ${\mathbb{F}}_q^n$ is asymptotically (as $n\to \infty$) at least the number expected in a random two-coloring. The linear equation is Sidorenko if the number of solutions in any dense subset of ${\mathbb{F}}_q^n$ is asymptotically at least the number expected in a random set of the same density. In this paper, we characterize those linear equations which are common, and those which are Sidorenko. The main novelty is a construction based on choosing random Fourier coefficients that shows that certain linear equations do not have these properties. This solves problems posed in a paper of Saad and Wolf.

11 pages