Sidorenko-Type Inequalities for Even Subdivisions over Finite Abelian Groups

Sidorenko's conjecture asserts that every bipartite graph $H$ has the property that, for any host graph $G$, the homomorphism density from $H$ to $G$ is asymptotically at least as large as in a quasirandom graph with the same edge density as $G$. While the conjecture remains still very open, Szegedy showed that it suffices to verify the inequality when the host graph is a Cayley graph over a finite group. In this paper, we prove that Sidorenko's conjecture holds for all even subdivisions of arbitrary graphs when the host graph is a Cayley graph over an abelian group. That is, if each edge of a graph is replaced by a path of even length (allowing different lengths for different edges), then the resulting graph satisfies the Sidorenko's inequality in any abelian Cayley host graph. Our approach reduces the homomorphism count to the evaluation of certain averages over solution sets of linear systems over finite abelian groups, and proceeds using Fourier-analytic techniques.