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. Sós, More on quasi-random graphs, subgraph counts and graph limits, arXiv:1405.6808].