Recurrence and primitivity for IP systems with polynomial wildcards

Abstract:

The IP Szemerédi Theorem of Furstenberg and Katznelson guarantees that for any positive density subset $E$ of a countable abelian group $G$ and for any sequences $(g_i^{(j)})_{i=1}^\infty$ in $G$, $1\leq j\leq k$, there is a finite non-empty $\alpha \subset {\mathbb {N}}$ such that $\bigcap _{j=1}^k ( E- \sum _{i\in \alpha } g_i^{(j)}) \neq \emptyset$. A natural question is whether, in this theorem, one may restrict $|\alpha |$ to, for example, the set $\{ n^d: d\in {\mathbb {N}}\}$. As a first step toward achieving this result, we develop here a new method for taking weak IP limits and prove a relevant projection theorem for unitary operators, which establishes as a corollary the case $k=2$ of the target result.