Common and Sidorenko Linear Equations

TL;DR

This paper characterizes which linear equations over finite fields are common or Sidorenko, using a novel Fourier coefficient construction to identify equations lacking these properties, thus solving open problems.

Contribution

It provides a complete characterization of common and Sidorenko linear equations over finite fields, introducing a new Fourier-based construction to demonstrate non-properties.

Findings

Identifies which linear equations are common in finite fields.

Identifies which linear equations are Sidorenko in finite fields.

Introduces a Fourier coefficient construction to show non-properties.

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.