Common and Sidorenko equations in Abelian groups

TL;DR

This paper investigates when linear configurations defined by single homogeneous equations are common in large finite Abelian groups, extending previous results from vector spaces over finite fields to broader group classes.

Contribution

It generalizes the characterization of common equations from finite fields to all sufficiently large Abelian groups with coprime coefficients.

Findings

Confirmed the conjecture for all large Abelian groups with coprime coefficients.

Extended the known results from vector spaces over finite fields to general Abelian groups.

Provided a complete classification of common equations in the specified group classes.

Abstract

A linear configuration is said to be common in a finite Abelian group $G$ if for every 2-coloring of $G$ the number of monochromatic instances of the configuration is at least as large as for a randomly chosen coloring. Saad and Wolf conjectured that if a configuration is defined as the solution set of a single homogeneous equation over $G$, then it is common in $\mathbb{F}_p^n$ if and only if the equation's coefficients can be partitioned into pairs that sum to zero mod $p$. This was proven by Fox, Pham and Zhao for sufficiently large $n$. We generalize their result to all sufficiently large Abelian groups $G$ for which the equation's coefficients are coprime to $\vert G\vert$