Arithmetic expanders and deviation bounds for random tensors

TL;DR

This paper establishes hypergraph variants of the Alon-Roichman theorem, providing new deviation bounds for sums of random multilinear forms and analyzing spectral expansion properties of Cayley hypergraphs.

Contribution

It introduces hypergraph spectral expansion variants, new deviation bounds for multilinear forms, and applies the polynomial method to hypergraph expansion analysis.

Findings

Existence of small sets of directions with controlled line distributions in finite fields

New deviation bounds for sums of independent random multilinear forms

Lower bounds on the size of generating sets for spectral expansion in hypergraphs

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 $p\geq 3$ and any $\varepsilon > 0$, there exists a set of directions $D\subseteq \mathbb{F}_p^n$ of size $O_{p,\varepsilon}(p^{(1-1/p +o(1))n})$ 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 $\varepsilon\alpha$ 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 $\varepsilon$-nets. Using the polynomial method we prove that a Cayley hypergraph with edges generated by a set~$D$ as above requires $|D| \geq \Omega_p(n^{p-1})$ for (our notion of) spectral expansion for hypergraphs.