Geodesic
Densities, submeasures and partitions of groups
Algebra and discrete mathematics, Tome 17 (2014) no. 2, pp. 193-221.

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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 $|F|\le \max_{0$; $G=F\cdot\bigcup_{x\in E}xA_iA_i^{-1}x^{-1}$ 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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Taras Banakh; Igor Protasov; Sergiy Slobodianiuk. Densities, submeasures and partitions of groups. Algebra and discrete mathematics, Tome 17 (2014) no. 2, pp. 193-221. http://geodesic.mathdoc.fr/item/ADM_2014_17_2_a2/

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