The four moment theorem asserts, roughly speaking, that the joint
distribution of a small number of eigenvalues of a Wigner random matrix (when
measured at the scale of the mean eigenvalue spacing) depends only on the first
four moments of the entries of the matrix. In this paper, we extend the four
moment theorem to also cover the coefficients of the \emph{eigenvectors} of a
Wigner random matrix. A similar result (with different hypotheses) has been
proved recently by Knowles and Yin, using a different method.

We give a new bound on the probability that the random sum $\xi_1 v_1 +...+
\xi_n v_n$ belongs to a ball of fixed radius, where the $\xi_i$ are iid
Bernoulli random variables and the $v_i$ are vectors in $\R^d$. As an
application, we prove a conjecture of Frankl and F\"uredi (raised in 1988),
which can be seen as the high dimensional version of the classical
Littlewood-Offord-Erd\H os theorem.

A random $n$-lift of a base graph $G$ is its cover graph $H$ on the vertices
$[n]\times V(G)$, where for each edge $u v$ in $G$ there is an independent
uniform bijection $\pi$, and $H$ has all edges of the form $(i,u),(\pi(i),v)$.
A main motivation for studying lifts is understanding Ramanujan graphs, and
namely whether typical covers of such a graph are also Ramanujan.