测量组的积集现象
Product set phenomena for measured groups
摘要：
继Furstenberg和Reynner关于平稳平均的工作之后，本文将Di Nasso，Goldbring，Jin，Leth，Lupini和Mahlburg关于可数\textsc{amenable}群中乘积集的分段并合性的一些结果加强和推广到一般可数测度群.我们指出的产品之间的行为的“大”集刘维尔和非刘维尔测量组的几个根本区别。作为我们主要结果的一个（非常）特殊的情况，我们证明了如果$G$是一个有限秩的自由群，$A$和$B$是$G$的“球面大”子集，则存在一个有限集$F \subset G$，使得$AFB$是厚的。集合$F$的位置很奇怪，但似乎是必要的;事实上，我们可以产生\emph{left thick}集合$A, B \subset G$，使得$B$是“球形大”的，但$AB$是\emph{not}分段联会的。另一方面，如果$A$是球面大的，则$AA^{-1}$总是分段联会的\emph{and}左分段联会的。然而，与顺从性组发生的情况相反，$AA^{-1}$可能不能联合。同样的现象也发生在许多其他（甚至是顺从的，但非刘维尔）被测量的群体中。我们的证明是基于一些遍历理论的结果，关于固定的行动，应该是独立的利益。
英文摘要：
Following the works of Furstenberg and Glasner on stationary means, 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 \textsc{amenable} groups to general countable measured groups. We point out 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 \emph{left thick} sets $A, B \subset G$ such that $B$ is "spherically large", but $AB$ is \emph{not} piecewise syndetic. On the other hand, if $A$ is spherically large, then $AA^{-1}$ is always piecewise syndetic \emph{and} left piecewise 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.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
