On partitions of G-spaces and G-lattices

Abstract

Abstract. Given a G-space X and a non-trivial G-invariant ideal I of subsets of X, we prove that for every partition X = A 1 ∪ · · · ∪ An of X into n ≥ 2 pieces there is a piece A i of the partition and a finite set F ⊂ G of cardinality |F | ≤ φ(n + 1) := max 1<x<n+1∈ I} is the difference set of the set A i . Also we investigate the growth of the sequence φ(n) = max 1<x<nand show that ln φ(n + 1) = nW (ne) − 2n +where W (x) is the Lambert W-function, defined implicitly as W (x)e W (x) = x. This shows that φ(n) grows faster that any exponent a n but slower than the sequence n! of factorials. Problem 1.1. Is it true that for any partition G = A 1 ∪ · · · ∪ A n of a group G into n pieces there is a piece Motivation, principal problems and results ThisA simple measure-theoretic argument shows that the answer to this problem is affirmative for any amenable group G. So, the problem actually concerns non-amenable groups. Let us recall that a group G is amenable if it admits a left-invariant finitely additive probability measure µ : P(X) → [0, 1] defined on the Boolean algebra P(X) of all subsets of X. In Theorem 12.7 of [7] Protasov and Banakh gave a partial answer to Problem 1.1 proving that for any partition G = A 1 ∪ · · · ∪ A n of a group G into n pieces there is a piece A i of the partition such that G = F A i A −1 i for some finite set F ⊂ G of cardinality |F | ≤ 2 2 n−1 −1 . They also observed that the answer to Problem 1.1 is affirmative for n ≤ 2.In [6] Protasov considered an "idealized" version of Problem 1.1. A family I of subsets of a set X is called an ideal on X if for any sets A, B ∈ I and C ∈ P(X) we get A ∪ B ∈ I and A ∩ C ∈ I. An ideal I on X is trivial if X ∈ I. Now assume that X is a G-space (i.e., a set endowed with a left action of a group G) and I is a G-invariant ideal on X. The G-invariantness of the ideal I means that for every g ∈ G and A ∈ I the shift gA of the set A belongs to the ideal I. For a subset A ⊂ X let ∆(A) = {g ∈ G : gA ∩ A / ∈ I} be the I-difference set of A. In [6] Protasov asked the following modification of Problem 1.1. Problem 1.2. Let X be an infinite G-space and I be the ideal of finite subsets of X. Is it true that for any partition X = A 1 ∪ · · · ∪ A n of X there is a piece A i of the partition such that G = F · ∆(A i ) for some finite set F ⊂ G of cardinality |F | ≤ n?The answer to this problem is affirmative if X admits a G-invariant probability measure. Also the upper bound 2 2 n−1 −1 on |F | from Theorem 12.7 [7] generalizes to the "idealized" setting, see [4]. Let us observe that Problem 1.2 actually concerns partitions of the Boolean algebra P(X)/I, so it is natural to consider this problem in context of Boolean algebras or more generally, bounded lattices.By a lattice we understand a set X endowed with two commutative idempotent associative operations ∨, ∧ : X × X → X connected by the absorption law: x ∨ (x ∧ y) = x and x ∧ (x ∨ y) = x for all x, y ∈ X. Each lattice (X, ∨, ∧) carries a natural partial order ≤ in which x ≤ y iff x ∧ y = x iff x ∨ y = y. A lattice ...

A note on the combinatorial derivation of non-small sets

“…One can in fact prove a similar result, with the same bound on |F |, for ∆ I by using Lemma 5 inductively (See [4]). However Banakh, Ravsky and Slobodianiuk [3] were able to prove a a stronger result, replacing the bound 2 2 n−1 −1 with some function φ(n) which, whilst growing quicker than any exponential function, is eventually bounded by n!.…”

The covering number of the difference sets in partitions of G-spaces and groups

“…The paper [3] gives a survey of available partial solutions of Protasov's Problems 1 and 2. Here we mention the following result of Banakh, Ravsky and Slobodianiuk [1]. Theorem 1.…”