On partitions of G-spaces and G-lattices

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 $|F|\le \phi(n+1):=\max_{1<x<n+1}\frac{x^{n+1-x}-1}{x-1}$suchthat$G=F\cdot \Delta(A_i)$where$\Delta(A_i)={g\in G:gA_i\cap A_i\notin I}$ is the difference set of the set $A_i$. Also we investigate the growth of the sequence $\varphi (n)={\max }_{1<x<n}\frac{x^{n-x}-1}{x-1}$ and show that $\ln \phi(n)=nW(ne)-2n+\frac{n}{W(ne)}+\frac{W(ne)}{n}+O\big(\frac{\ln n}n\big)$where$W(x)$istheLambertW-function,definedimplicitlyas$W(x)e^{W(x)}=x$. This shows that $\varphi (n)$ grows faster that any exponent $a^n$ but slower than the sequence of factorials $n!$.