Densities, submeasures and partitions of groups

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.