Densities, submeasures and partitions of groups

Abstract

In 1995 in Kourovka notebook the second author asked the following problem: is it true that for each partition G=A₁ ∪ ⋯ ∪ An of a group G there is a cell Ai of the partition such that G = FAiA⁻¹i for some set F ⊂ G of cardinality |F |≤ n? In this paper we survey several partial solutions of this problem, in particular those involving certain canonical invariant densities and submeasures on groups. In particular, we show that for any partition G = A₁ ∪ ⋯ ∪ An of a group G there are cells Ai, Aj of the partition such that G = FAjA⁻¹j for some finite set F ⊂ G of cardinality |F| ≤ max₀<k≤n ∑ⁿ⁻kp₌₀kp ≤ n!; G = F ⋅ ⋃x∈ExAiA⁻¹ix⁻¹ for some finite sets F, E ⊂ G with |F| ≤ n; G = FAiA⁻¹iAi for some finite set F ⊂ G of cardinality |F| ≤ n; the set (AiA⁻¹i)⁴ⁿ⁻¹ is a subgroup of index ≤ n in G. The last three statements are derived from the corresponding density results.