Towards a characterisation of Sidorenko systems

TL;DR

This paper investigates the properties of Sidorenko systems of linear forms over finite fields, providing necessary conditions, identifying structured examples, and advancing towards a complete classification of such systems.

Contribution

It introduces a necessary condition for Sidorenko systems, constructs a large family of structured Sidorenko systems using the entropy method, and advances the classification of two-equation systems.

Findings

Identified a necessary condition for Sidorenko systems.

Constructed a large family of structured Sidorenko systems.

Made progress towards classifying systems of two equations.

Abstract

A system of linear forms $L=\{L_1,\ldots,L_m\}$ over $\mathbb{F}_q$ is said to be Sidorenko if the number of solutions to $L=0$ in any $A \subseteq \mathbb{F}_{q}^n$ is asymptotically as $n\to\infty$ at least the expected number of solutions in a random set of the same density. Work of Saad and Wolf (2017) and of Fox, Pham and Zhao (2019) fully characterises single equations with this property and both sets of authors ask about a characterisation of Sidorenko systems of equations. In this paper, we make progress towards this goal. Firstly, we find a simple necessary condition for a system to be Sidorenko, thus providing a rich family of non-Sidorenko systems. In the opposite direction, we find a large family of structured Sidorenko systems, by utilising the entropy method. We also make significant progress towards a full classification of systems of two equations.