Quotient Sets and Density Recurrent Sets

Let S be a left amenable semigroup. Say that a subset A of S is large if there is some left invariant mean μ on S with μ(χA) > 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 ∈ B such that x−1A ∩ A is large. We show that the set DR(S) of ultrafilters on S, every member of which is density recurrent, is a compact subsemigroup of the Stone-Čech compactification βS of S containing the idempotents of βS. If S is a group, we show that for every nonprincipal ultrafilter p on S, p−1p ∈ DR(S), where p−1 = {A−1 : A ∈ 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−1p for each k ∈ N. We show that DR(N,+) has substantial multiplicative structure. We show further that if A is a large subset of S, then DR(S) ⊆ AA−1, where the quotient set AA−1 = {x ∈ S : (∃y ∈ A)(xy ∈ A)}. For each positive integer n, we introduce the notion of a polynomial n-recurrent set in 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 (βN,+) containing the additive idempotents and a left ideal of (βN, ·).