Product set phenomena for measured groups

We strengthen and extend in this paper some recent results by Di Nasso, Goldbring, Jin, Leth, Lupini and Mahlburg on piecewise syndeticity of product sets in countable amenable groups to general countable measured groups. We also address several fundamental differences between the behavior of products of ‘large’ sets in Liouville and non-Liouville measured groups. As a (very) special case of our main results, we show that if $G$ is a free group of finite rank, and $A$ and $B$ are ‘spherically large’ subsets of $G$, then there exists a finite set $F\subset G$ such that $AFB$ is thick. The position of the set $F$ is curious, but seems to be necessary; in fact, we can produce left thick sets $A,B\subset G$ such that $B$ is ‘spherically large’, but $AB$ is not piecewise syndetic. On the other hand, if $A$ is spherically large, then $AA^{-1}$ is always piecewise syndetic and piecewise left syndetic. However, contrary to what happens for amenable groups, $AA^{-1}$ may fail to be syndetic. The same phenomena occur for many other (even amenable, but non-Liouville) measured groups. Our proofs are based on some ergodic-theoretical results concerning stationary actions which should be of independent interest.

HIGH DENSITY PIECEWISE SYNDETICITY OF PRODUCT SETS IN AMENABLE GROUPS

M. Beiglböck, V. Bergelson, and A. Fish proved that if G is a countable amenable group and A and B are subsets of G with positive Banach density, then the product set AB is piecewise syndetic. This means that there is a finite subset E of G such that EAB is thick, that is, EAB contains translates of any finite subset of G. When G = ℤ, this was first proven by R. Jin. We prove a quantitative version of the aforementioned result by providing a lower bound on the density (with respect to a Følner sequence) of the set of witnesses to the thickness of EAB. When G = ℤd, this result was first proven by the current set of authors using completely different techniques.