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 {x₁,…,x_k}∈(Gk) with x₁+…+x_k∈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=Z_n.