arXiv subjects
Publications and source records attributed to Sergiy Slobodianiuk.
In 1995 in Kourovka notebook the second author asked the following problem: it is 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.
We prove that for every number k each countable infinite group $G$ admits a partition $G=A\cup B$ into two sets which are $k$-meager in the sense that for every $k$-element subset $K\subset G$ the sets $KA$ and $KB$ are not thick. The proof is based on the fact that $G$ possesses a syndetic submeasure, i.e., a left-invariant submeasure $μ:\mathcal P(G)\to[0,1]$ such that for each $ε> 1/|G|$ and subset $A\subset G$ with $μ(A)<1$ there is a set $B\subset G\setminus A$ such that $μ(B)<ε$ and $FB=G$ for some finite subset $F\subset G$.
Given a $G$-space $X$ and a non-trivial $G$-invariant ideal $I$ of subsets of $X$, we prove that for every partition $X=A_1\cup\dots\cup A_n$ of $X$ into $n\ge 2$ pieces there is a piece $A_i$ of the partition and a finite set $F\subset G$ of cardinality $|F|\le ϕ(n+1):=\max_{1<x<n+1}\frac{x^{n+1-x}-1}{x-1}$ such that $G=F\cdot Δ(A_i)$ where $Δ(A_i)=\{g\in G:gA_i\cap A_i\notin I\}$ is the difference set of the set $A_i$. Also we investigate the growth of the sequence $ϕ(n)=\max_{1<x<n}\frac{x^{n-x}-1}{x-1}$ and show that $\ln ϕ(n)=nW(ne)-2n+\frac{n}{W(ne)}+\frac{W(ne)}{n}+O\big(\frac{\ln n}n\big)$ where $W(x)$ is the Lambert W-function, defined implicitly as $W(x)e^{W(x)}=x$. This shows that $ϕ(n)$ grows faster that any exponent $a^n$ but slower than the sequence of factorials $n!$.