Multiple recurrence and nilsequences
Résumé
Aiming at a simultaneous extension of Khintchine's and
Furstenberg's Recurrence theorems, we address the question if
for a measure
preserving system $(X,\CX,\mu,T)$ and a set $A\in\CX$ of positive
measure, the set of integers $n$ such that $\mu(A\cap
T^nA\cap T^{2n}A\cap \ldots\cap T^{kn}A)
>\mu(A)^{k+1}-\epsilon$ is syndetic.
The size of this set, surprisingly
enough, depends on the length $(k+1)$ of the arithmetic progression
under consideration. In an ergodic system, for $k=2$ and $k=3$,
this set is syndetic, while for $k\geq 4$ it is not.
The main tool is a decomposition result for the
multicorrelation sequence $\int f(x)f(T^nx)f(T^{2n}x)\ldots f(T^{kn}x)
\,d\mu(x)$, where $k$ and $n$ are positive integers and
$f$ is a bounded measurable function.
We also derive combinatorial consequences of these results,
for example showing that for a set of integers $E$ with upper
Banach density $d^*(E)>0$ and for all $\epsilon > 0$, the set
$$
\{n\in\Z\colon d^*\bigl(E\cap (E+n)\cap (E+2n)\cap (E+3n)\bigr)>
d^*(E)^4-\epsilon\}$$
is syndetic.