Syndetic Submeasures and Partitions of G-Spaces and Groups

Abstract

We prove that for every k ∈ N each countable infinite group G admits a partition G = A ∪ B into two sets which are k-meager in the sense that for every k-element subset K ⊂ G the sets KA and KB are not thick. The proof is based on the fact that G possesses a syndetic submeasure, i.e., a left-invariant submeasure µ : P(G) → [0, 1] such that for each ε > 1

Note: A conjecture on partitions of groups

“…It is still an open problem posed in [6,Problem 13.44] whether Φ(n) = n. For the history and results behind this problem see the survey [1].…”

Partitions of Groups

“…This is Problem 13.44 from the Kourovka notebook [23] posed by the first author in 1995. For nowday state of this open problem see the survey [6]. Here we formulate some results from [6].…”

“…For every countable group, the negative answer were obtained in [7] with help of syndedic submeasures.…”

A note-question on partitions of semigroups

“…For the current state of this open problem see the survey [1]. We mention only that an answer is positive if either G is amenable (in particular, finite), or n ≤ 3, or x −1 Ax = A for any A ∈ P and x ∈ G. If G is an arbitrary group and P is an n-partition of G then one can choose A, B ∈ P and subsets F , H of G such that G = F AA −1 , |F | ≤ n!…”