Quotient sets and density recurrent sets

Abstract:

Let $S$ be a left amenable semigroup. Say that a subset $A$ of $S$ is large if there is some left invariant mean $\mu$ on $S$ with $\mu (\chi _A)>0$. A subset $B$ of $S$ is density recurrent if and only if, whenever $A$ is a large subset of $S$, there is some $x\in B$ such that $x^{-1}A\cap A$ is large. We show that the set $\mathcal {DR}(S)$ of ultrafilters on $S$, every member of which is density recurrent, is a compact subsemigroup of the Stone-Čech compactification $\beta S$ of $S$ containing the idempotents of $\beta S$. If $S$ is a group, we show that for every nonprincipal ultrafilter $p$ on $S$, $p^{-1} p\in \mathcal {DR}(S)$, where $p^{-1}=\{A^{-1}:A\in p\}$. We obtain combinatorial characterizations of sets which are members of a product of $k$ idempotents and of sets which are members of a product of $k$ elements of the form $p^{-1} p$ for each $k\in \mathbb {N}$. We show that $\mathcal {DR}(\mathbb {N},+)$ has substantial multiplicative structure. We show further that if $A$ is a large subset of $S$, then $\mathcal {DR}(S)\subseteq \overline {AA^{-1}}$, where the quotient set $AA^{-1}=\{x\in S:(\exists y\in A)(xy\in A)\}$. For each positive integer $n$, we introduce the notion of a polynomial $n$-recurrent set in $\mathbb {N}$. (Such sets provide a generalization of the polynomial Szemerédi Theorem.) We show that the ultrafilters, every member of which is a polynomial $n$-recurrent set, are a subsemigroup of $(\beta \mathbb {N},+)$ containing the additive idempotents and a left ideal of $(\beta \mathbb {N},\cdot )$.