Common and Sidorenko equations in Abelian groups

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 in an even number of variables over $G$, then it is common in $\mathbb{F}^n_p$ if and only if the equation’s coefficients can be partitioned into pairs that sum to zero $\operatorname{mod} p$. 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 $\lvert G \rvert$.

Keywords

linear configuration, Sidorenko, common, Abelian group

2010 Mathematics Subject Classification

Primary 11B75. Secondary 05D10.

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The author is grateful to be funded by Trinity College of the University of Cambridge through the Trinity External Researcher Studentship.

Received 20 September 2021

Accepted 30 November 2021

Published 19 August 2022