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

TL;DR

This paper characterizes integrable functions on the unit cube that have zero average over products of disjoint sets with specified measures, providing insights into quasi-random graph properties and settling related conjectures.

Contribution

It offers a complete characterization of functions with vanishing averages over products of disjoint sets, advancing understanding in quasi-random graph theory.

Findings

Characterization of functions with zero averages over disjoint set products

Resolution of specific conjectures in quasi-random graph theory

Enhanced understanding of functions related to graph limits

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].