Arithmetic expanders and deviation bounds for random tensors

Abstract

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 prime $p\ge 3$ and any $\epsilon >0$, there exists a set of directions $D\subseteq {\mathbb{F}}_p^n$ of size $O_{p,\epsilon }(p^{(1-1/p+o(1))n})$ such that for every set $A\subseteq {\mathbb{F}}_p^n$ of density $\alpha$, the fraction of lines in $A$ with direction in $D$ is within $\epsilon \alpha$ 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 $\epsilon$-nets. Using the polynomial method we prove that a Cayley hypergraph with edges generated by a set~$D$ as above requires $|D|\ge {\Omega }_p(n^{p-1})$ for (our notion of) spectral expansion for hypergraphs.

References in corpus (5)

• The dichotomy between structure and randomness, arithmetic progressions, and the primes
• Translation invariant equations and the method of Sanders
• On large subsets of $F_q^n$ with no three-term arithmetic progression
• Quasirandom Cayley graphs
• Locally decodable codes and the failure of cotype for projective tensor products