Linear Quasi-Randomness of Subsets of Abelian Groups and Hypergraphs

We establish an equivalence between the two seemingly distant notions of quasi-randomness: small linear bias of subsets of abelian groups and uniform edge distribution for uniform hypergraphs. For a subset A⊂G of an abelian group G consider the k-uniform Cayley (sum) hypergraph H(k)(A). The vertex set of H(k)(A) is G and the edges are k-element sets [Fourmula presented]. For d∈(0,1) we show that sets A⊂G of density d+o(1) have all non-trivial Fourier coefficients of order o(|G|) if and only if [Fourmula presented] for all U⊂V(H(k)(A)). This connects the work of Chung and Graham on quasi-random subsets of the integers and that of Conlon–Hàn–Person–Schacht on weak/linear quasi-random hypergraphs. Further, it extends the work of Chung and Graham who established the corresponding result for k=2 and G=Zn. © 2020 Elsevier Ltd