On partitions of G-spaces and G-lattices

Abstract

Given a $G$-space $X$ and a non-trivial $G$-invariant ideal $I$ of subsets of $X$, we prove that for every partition $X=A_1\cup \cdots \cup A_n$ 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 $|F|\le \varphi (n+1):={\max }_{1<x<n+1}\frac{x^{n+1-x}-1}{x-1}$ such that $G=F\cdot \Delta (A_i)$ where $\Delta (A_i)=\{g\in G:g{A}_i\cap A_i\notin I\}$ is the difference set of the set $A_i$. Also we investigate the growth of the sequence $\varphi (n)={\max }_{1<x<n}\frac{x^{n-x}-1}{x-1}$ and show that $\ln \varphi (n)=nW(ne)-2n+\frac{n}{W(ne)}+\frac{W(ne)}{n}+O\left(\frac{\ln n}{n}\right)$ where $W(x)$ is the Lambert W-function, defined implicitly as $W(x)e^{W(x)}=x$. This shows that $\varphi (n)$ grows faster that any exponent $a^n$ but slower than the sequence of factorials $n!$.

21 pages + 12 pages in Appendix