On a question of Alon

TL;DR

This paper investigates the property of linear systems in finite fields being 'common' under two-colorings, disproving a conjecture that adding free variables can make any system uncommon, and explores related notions of commonness.

Contribution

It provides a negative answer to Alon's question about the effect of adding free variables on the commonness of linear systems, and connects this to geometric mean-based notions of commonness.

Findings

Disproved that adding free variables makes any system uncommon.

Connected commonness to geometric mean-based solution counts.

Resolved related questions by Kamčev–Liebenau–Morrison.

Abstract

A system of linear equations in $\mathbb{F}_p^n$ is \textit{common} if every two-colouring of $\mathbb{F}_p^n$ yields at least as many monochromatic solutions as a random two-colouring, asymptotically as $n \to \infty$. By analogy to the graph-theoretic setting, Alon has asked whether any (non-Sidorenko) system of linear equations can be made uncommon by adding sufficiently many free variables. Fox, Pham and Zhao answered this question in the affirmative among systems which consist of a single equation. We answer Alon's question in the negative. We also observe that the property of remaining common despite that addition of arbitrarily many free variables is closely related to a notion of commonness in which one replaces the arithmetic mean of the number of monochromatic solutions with the geometric mean, and furthermore resolve questions of Kam\v{c}ev--Liebenau--Morrison.