Towards a characterization of Sidorenko systems

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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 [19] and of Fox, Pham and Zhao [8] fully characterizes single equations with this property and both sets of authors ask about a characterization of Sidorenko systems of equations. In this paper, we make progress towards this goal. First, 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 utilizing the entropy method. We also make significant progress towards a full classification of systems of two equations.
Funding Information
  • Australian Research Council Discovery Project (DP180103684)
  • Marie Sklodowska-Curie grant agreement (01038085)
  • Natural Sciences and Engineering Research Council of Canada (RGPIN-2021-02511)

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