On partitions of G-spaces and G-lattices

TL;DR

This paper proves that for any partition of a G-space with a G-invariant ideal, one piece's difference set combined with a small finite set covers the group, and analyzes the growth rate of a related function involving the Lambert W-function.

Contribution

It establishes a new partition property for G-spaces involving difference sets and provides an asymptotic analysis of the growth of a related combinatorial function.

Findings

Existence of a finite set F with bounded size such that G=F·Δ(A_i) for some partition piece A_i.

The growth rate of φ(n) exceeds any exponential but is slower than factorials.

Asymptotic formula for ln φ(n) involving Lambert W-function.

Abstract

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