Low discrepancy and spectral gap for directed hypergraphs via spectral regularity lemma for tensors

Abstract

We show that low discrepancy is equivalent to spectral gap for directed $k$-uniform hypergraphs. For undirected hypergraphs this recovers a theorem of Lenz and Mubayi, with a considerably shorter proof. At the heart of our argument is a Frieze--Kannan type spectral regularity lemma for hypergraphs based on the variational notion of hypergraph eigenvalues by Friedman--Wigderson, which decomposes any tensor into a bounded number of rank-one tensors plus a quasi-random tensor with small top eigenvalue. This lemma may be of independent interest, and we prove it for complex-valued, not necessarily symmetric tensors. We also briefly discuss a Szemer\'edi-type variant and the regularization of Cayley-type hypergraphs over finite abelian groups.