Largeness of the set of finite products in a semigroup

A history of central sets

“…Let M = F∈I C F . By condition (i) and [53,Theorem 4.20], M is a subsemigroup of β S. By assumption (1) and [53,Theorem 3.11], M ∩ T = ∅ and thus M ∩ T is a compact semigroup which therefore has an idempotent p. Since M ⊆ A, we have that A is a Z -set.…”

Notions of size in a semigroup: an update from a historical perspective

“…To see that q p ∈ Prog(S), let A ∈ q p and let k ∈ N. We show that A contains a length k progression. Now A ∈ λ q ( p) and λ q is continuous with respect to so pick (2) and note that ρ x(t) (q) ∈ A so that we may pick…”

New examples of complete sets, with connections to a Diophantine theorem of Furstenberg

“…Then F +S ′ +log a b n ℓ α is ε-dense in [0, ∞). Since the exponential function x → a x is 2 log(a)-Lipschitz on (−∞, log a (2)], a S ′ b {0,...,k} b n ℓ α is 2 log(a)ε-dense in [1,2]. But this implies that A ′ α ⊇ a S ′ b {n ℓ ,...,n ℓ +k} α is 2 log(b)ε-dense in T. Since ε was arbitrary, this completes the proof.…”

“…[12,3], and see [2] for a generalization). 1 However, the differences between these definitions are significant, and we will not discuss Glasner sets in this paper.…”