  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.
