Large Sets in Boolean and Non-Boolean Groups and Topology

Abstract

Various notions of large sets in groups, including the classical notions of thick, syndetic, and piecewise syndetic sets and the new notion of vast sets in groups, are studied with emphasis on the interplay between such sets in Boolean groups. Natural topologies closely related to vast sets are considered; as a byproduct, interesting relations between vast sets and ultrafilters are revealed.Keywords: large set in a group; vast set; syndetic set; thick set; piecewise syndetic set; Boolean topological group; arrow ultrafilter; Ramsey ultrafilter Various notions of large sets in groups and semigroups naturally arise in dynamics and combinatorial number theory. Most familiar are those of syndetic, thick (or replete), and piecewise syndetic sets. Apparently, the term "syndetic" was introduced by Gottschalk and Hedlund in their 1955 book [1] in the context of topological groups, although syndetic sets of integers have been studied long before (they appear, e.g., in Khintchine's 1934 ergodic theorem). During the past decades, large sets in Z and in abstract semigroups have been extensively studied. It has turned out that, e.g., piecewise syndetic sets in N have many attractive properties: they are partition regular (i.e., given any partition of N into finitely many subsets, at least one of the subsets is piecewise syndetic), contain arbitrarily long arithmetic progressions, and are characterized in terms of ultrafilters on N (namely, a set is piecewise syndetic it and only if it belongs to an ultrafilter contained in the minimal two-sided ideal of βN). Large sets of other kinds are no less interesting, and they have numerous applications to dynamics, Ramsey theory, the ultrafilter semigroup on N, the Bohr compactification, and so on.Quite recently Reznichenko and the author have found yet another application of large sets. Namely, we introduced special large sets in groups, which we called vast, and applied them to construct a discrete set with precisely one limit point in any countable nondiscrete topological group in which the identity element has nonrapid filter of neighborhoods. Using this technique and special features of Boolean groups, we proved, in particular, the nonexistence of a countable nondiscrete extremally disconnected group in ZFC (see [2]).In this paper, we study right and left thick, syndetic, piecewise syndetic, and vast sets in groups (although they can be defined for arbitrary semigroups). Our main concern is the interplay between such sets in Boolean groups. We also consider natural topologies closely related to vast sets, which leads to interesting relations between vast sets and ultrafilters. Basic Definitions and NotationWe use the standard notation Z for the group of integers, N for the set (or semigroup, depending on the context) of positive integers, and ω for the set of nonnegative integers or the first infinite cardinal; we identify cardinals with the corresponding initial ordinals. Given a set X, by |X| we denote its cardinality, by [X] k for k ∈ N, the kth symmetric power of ...

Ramsey-product subsets of a group

“…The second statement is proved in Example 4 of [7] and the third statement trivially follows from the second statement. Now we endow G with the discrete topology and identify the Stone-Čech compactification βG of G with the set of all ultrafilters on G. The family {Ā : A ⊆ G}, whereĀ = {p ∈ βG : A ∈ p}, forms the base for the topology of βG.…”

“…In particular, Ramsey (−1, 1)-product sets are exactly ∆ ω sets from [6], Ramsey (−1, 1)-product sets containing the unit of the group are exactly ω-fat sets of Sipacheva [7]; Ramsey (−1, 1)-product sets are similar to ∆ * -sets, studied in [1] (see [7, p.6]). Now we present some examples and establish some topological properties of Ramsey ⃗ m-product sets.…”