Towards a characterisation of Sidorenko systems

A system of linear forms L = {L1, . . . , Lm} over Fq is said to be Sidorenko if the number of solutions to L = 0 in any A ⊆ Fq is asymptotically as n → ∞ at least the expected number of solutions in a random set of the same density. Work of Saad and Wolf [18] and of Fox, Pham and Zhao [8] 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 utilizing the entropy method. We also make significant progress towards a full classification of systems of two equations.

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