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 \(G=A_1\cup\dots\cup A_n\) 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.