Arithmetic Expanders and Deviation Bounds for Sums of Random Tensors

We prove hypergraph variants of the celebrated Alon–Roichman theorem on spectral expansion of sparse random Cayley graphs. One of these variants implies that for every odd prime p and any ε > 0, there exists a set of directions D ⊆ Fp of size Op,ε(p (1−1/p+o(1))n) such that for every set A ⊆ Fp of density α, the fraction of lines in A with direction in D is within εα of the fraction of all lines in A. Our proof uses new deviation bounds for sums of independent random multi-linear forms taking values in a generalization of the Birkhoff polytope. The proof of our deviation bound is based on Dudley’s integral inequality and a probabilistic construction of ε-nets. Using the polynomial method we prove that a Cayley hypergraph with edges generated by a set D as above requires |D| ≥ Ωp(n) for (our notion of) spectral expansion for hypergraphs.