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 C 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 {xi,, xk} E (Gk) with xi xk E A. Ford E (0, 1) we show that sets A C G of density d o(1) have all non-trivial Fourier coefficients of order o(IGI) if and only if e(U) = drkl) o(IG1k) for all U c V(H(k)(A)). This connects the work of Chung and Graham on quasi -random subsets of the integers and that of Conlon -Han 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. (C) 2020 Elsevier Ltd. All rights reserved.