Linear quasi-randomness of subsets of abelian groups and hypergraphs

Abstract

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 {x1,…,xk}∈(Gk) with x1+…+xk∈A. 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 e(U)=d(|U|k)+o(|G|k) 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.