A characterization of functions with vanishing averages over products of disjoint sets

Abstract

Given $\alpha_1,...,\alpha_m \in (0,1)$, we characterize all integrable functions $f:[0,1]^m \to \mathbb{C}$ satisfying $\int_{A_1 \times ...\times A_m} f =0$ for any collection of disjoint sets $A_1,...,A_m \subseteq [0,1]$ of respective measures $\alpha_1,...,\alpha_m$. We use this characterization to settle some of the conjectures in [S. Janson and V. S\'os, More on quasi-random graphs, subgraph counts and graph limits, arXiv:1405.6808].