Towards a characterisation of Sidorenko systems

A system of linear forms $L=\{L_1,\ldots,L_m\}$ over $\mathcal{F}_q$ is said to be \emph{Sidorenko} if the number of solutions to $L=0$ in any $A \subseteq \mathcal{F}_{q}^n$ 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~\cite{sw17} and of Fox, Pham and Zhao~\cite{fpz19} fully characterises single equations with this property and both sets of authors ask about a characterisation of Sidorenko \emph{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.