### Densities, submeasures and partitions of groups

Taras Banakh, Igor Protasov, Sergiy Slobodianiuk

#### Abstract

In 1995 in Kourovka notebook the second author asked the following problem:  is it true that for each partition $$G=A_1\cup\dots\cup A_n$$ of a group $$G$$ there is a cell $$A_i$$ of the partition such that $$G=FA_iA_i^{-1}$$ for some set $$F\subset G$$ of cardinality $$|F|\le n$$?  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 $$G=A_1\cup\dots\cup A_n$$  of a group $$G$$ there are cells $$A_i$$, $$A_j$$ of the partition such that

$$G=FA_jA_j^{-1}$$ for some finite set $$F\subset G$$ of cardinality $$|F|\le \max_{0<k\le n}\sum_{p=0}^{n-k}k^p\le n!$$;

$$G=F\cdot\bigcup_{x\in E}xA_iA_i^{-1}x^{-1}$$ for some finite sets $$F,E\subset G$$ with $$|F|\le n$$;

$$G=FA_iA_i^{-1}A_i$$ for some finite set $$F\subset G$$ of cardinality $$|F|\le n$$;

the set $$(A_iA_i^{-1})^{4^{n-1}}$$ is a subgroup of index $$\le n$$ in $$G$$.

The last three statements are derived from the corresponding density results.

#### Keywords

partition of a group; density; submeasure; amenable group

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