Non-conventional ergodic averages and nilmanifolds
Abstract
We study the $L^2$-convergence of two types of ergodic averages.
The first is the average of a product of functions evaluated at return times
along arithmetic progressions, such as the expressions appearing in
Furstenberg's proof of Szemer\'edi's Theorem. The second average is taken
along cubes whose sizes tend to
$+\infty$.
For each average, we show that it is sufficient to prove the
convergence for special systems, the \emph{characteristic factors}.
We build these factors in a general way, independent of the type of
the average. To each of these factors we associate a natural group of
transformations and give them the structure of a nilmanifold.
From the second convergence result we derive a combinatorial interpretation
for the arithmetic structure inside a set of integers of
positive upper density.