Linear quasi-randomness of subsets of abelian groups and hypergraphs

Quasi-random properties are deterministic properties which capture certain characteristics of random objects. The last decades have seen an extensive e↵ort in the study of this subject which has revealed many connections between di↵erent branches of mathematics and theoretical computer science. For graphs the systematic investigation was initiated by Thomason [27, 28] who studied deterministic graph sequences which exhibit a key property of the binomial random graph G(n, p): uniform edge distribution. Subsequently, in a cornerstone result of the area [8], Chung, Graham and Wilson proved that many graph properties characteristic for G(n, p) are indeed equivalent to a qualitative version of uniform edge distribution. For example, a graph sequence has uniform edge distribution if and only if the number of labelled copies of each fixed graph is about what is expected from the binomial random graph with the same edge density. We refer to [8] for the full statement of the result and to [23] for a survey on the subject.