Quotient sets and density recurrent sets (Q2841360)

This paper has a variety of results concerning subsets of a left amenable semigroup \(S\). A subset \(A\subset S\) is deemed to be large if it is given positive measure by some left-invariant mean on \(S\), and is density recurrent if, for any large set \(A\), it contains some \(x\in S\) for which \(x^{-1}A\cap A\) is also large. The first result is that the collection \(\mathrm{DR}(S)\) of those ultrafilters on \(S\) every member of which is density recurrent is a compact subsemigroup of \(\beta S\); in the case that \(S\) is a group, it is shown that \(p^{-1}p\in \mathrm{DR}(S)\) if \(p\) is nonprincipal ultrafilter. Those sets which are members of a product of \(k\) idempotents, or a product of \(k\) elements of the form \(p^{-1}p\), are characterized combinatorially. Several other results on combinatorial and algebraic properties of large sets and of \(\mathrm{DR}(S)\) are given. In particular, they show that those ultrafilters every member of which is a polynomial \(n\)-recurrent set form a subsemigroup of \((\beta\mathbb N, +)\) containing the additive idempotents and is a left ideal of \((\mathbb N,\cdot)\).