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 introduce a randomized iterative fragmentation procedure for finite metric
spaces, which is guaranteed to result in a polynomially large subset that is
$D$-equivalent to an ultrametric, where $D\in (2,\infty)$ is a prescribed
target distortion. Since this procedure works for $D$ arbitrarily close to the
nonlinear Dvoretzky phase transition at distortion 2, we thus obtain a much
simpler probabilistic proof of the main result of Bartel, Linial, Mendel, and
Naor, answering a question from Mendel and Naor, and yielding the best known
bounds in the nonlinear Dvoretzky theorem.

Using the density-increment strategy of Roth and Gowers, we derive
Szemeredi's theorem on arithmetic progressions from the inverse conjectures
GI(s) for the Gowers norms, recently established by the authors and Ziegler.

The Furstenberg recurrence theorem (or equivalently, Szemer\'edi's theorem)
can be formulated in the language of von Neumann algebras as follows: given an
integer $k \geq 2$, an abelian finite von Neumann algebra $(\M,\tau)$ with an
automorphism $\alpha: \M \to \M$, and a non-negative $a \in \M$ with
$\tau(a)>0$, one has $\liminf_{N \to \infty} \frac{1}{N} \sum_{n=1}^N \Re
\tau(a \alpha^n (a) ... \alpha^{(k-1)n} (a)) > 0$; a subsequent result of Host
and Kra shows that this limit exists. In particular, $\Re \tau(a \alpha^n (a)
>...

Let $(X,d,\mu)$ be a metric measure space. For $\emptyset\neq R\subseteq
(0,\infty)$ consider the Hardy-Littlewood maximal operator $$ M_R f(x)
\stackrel{\mathrm{def}}{=} \sup_{r \in R} \frac{1}{\mu(B(x,r))} \int_{B(x,r)}
|f| d\mu.$$ We show that if there is an $n>1$ such that one has the
"microdoubling condition" $ \mu(B(x,(1+\frac{1}{n})r))\lesssim \mu(B(x,r)) $
for all $x\in X$ and $r>0$, then the weak $(1,1)$ norm of $M_R$ has the
following localization property: $$ \|M_R\|_{L_1(X) \to L_{1,\infty}(X)}\asymp
\sup_{r>0} \|M_{R\cap [r,nr]}\|_{L_1(X) \to L_{1,\infty}(X)}.

The inverse conjecture for the Gowers norms $U^d(V)$ for finite-dimensional
vector spaces $V$ over a finite field $\F$ asserts, roughly speaking, that a
bounded function $f$ has large Gowers norm $\|f\|_{U^d(V)}$ if and only if it
correlates with a phase polynomial $\phi = e_\F(P)$ of degree at most $d-1$,
thus $P: V \to \F$ is a polynomial of degree at most $d-1$.

Let $G = (G,+)$ be an additive group. The sumset theory of Pl\"unnecke and
Ruzsa gives several relations between the size of sumsets $A+B$ of finite sets
$A, B$, and related objects such as iterated sumsets $kA$ and difference sets
$A-B$, while the inverse sumset theory of Freiman, Ruzsa, and others
characterises those finite sets $A$ for which $A+A$ is small. In this paper we
establish analogous results in which the finite set $A \subset G$ is replaced
by a discrete random variable $X$ taking values in $G$, and the cardinality
$|A|$ is replaced by the Shannon entropy $\Ent(X)$.

We strengthen the Carleson-Hunt theorem by proving $L^p$ estimates for the
$r$-variation of the partial sum operators for Fourier series and integrals,
for $p>\max\{r',2\}$. Four appendices are concerned with transference, a
variation norm Menshov-Paley-Zygmund theorem, and applications to nonlinear
Fourier transforms and ergodic theory.

Let $<\P > \subset \N$ be a multiplicative subsemigroup of the natural
numbers $\N = \{1,2,3,...\}$ generated by an arbitrary set $\P$ of primes
(finite or infinite). We given an elementary proof that the partial sums
$\sum_{n \in < \P >: n \leq x} \frac{\mu(n)}{n}$ are bounded in magnitude by 1.
With the aid of the prime number theorem, we also show that these sums converge
to $\prod_{p \in \P} (1 - \frac{1}{p})$ (the case when $\P$ is all the primes
is a well-known observation of Landau).