Difference bases in finite Abelian groups

Abstract

A subset $B$ of a group $G$ is called a difference basis of $G$ if each element $g\in G$ can be written as the difference $g=a{b}^{-1}$ of some elements $a,b\in B$. The smallest cardinality $|B|$ of a difference basis $B\subset G$ is called the difference size of $G$ and is denoted by $\Delta [G]$. The fraction $ð[G]:=\frac{\Delta [G]}{\sqrt{|G|}}$ is called the difference characteristic of $G$. Using properies of the Galois rings, we prove recursive upper bounds for the difference sizes and characteristics of finite Abelian groups. In particular, we prove that for a prime number $p\ge 11$, any finite Abelian $p$-group $G$ has difference characteristic $ð[G]<\frac{\sqrt{p}-1}{\sqrt{p}-3}\cdot {\sup }_{k\in \mathbb{N}}ð[C_{p^k}]<\sqrt{2}\cdot \frac{\sqrt{p}-1}{\sqrt{p}-3}$. Also we calculate the difference sizes of all Abelian groups of cardinality $<96$.

12 pages