Unique Difference Bases of Z

For $n\in \mathbb{Z}, A\subset \mathbb{Z}$ , let $\delta_{A}(n)$ denote the number of representations of

in the form

, where $a,a'\in A$ . A set $A\subset \mathbb{Z}$ is called a unique difference basis of $\mathbb{Z}$ if $\delta_{A}(n)=1$ for all $n\neq 0$ in $\mathbb{Z}$ . In this paper, we prove that there exists a unique difference basis of $\mathbb{Z}$ whose growth is logarithmic. These results show that the analogue of the Erdos-Turán conjecture fails to hold in $(\mathbb{Z},-)$ .

Received October 12 2010; revised version received January 26 2011. Published in Journal of Integer Sequences, February 9 2011.

Unique Difference Bases of Z

Chi-Wu Tang, Min Tang, and Lei Wu Department of Mathematics Anhui Normal University Wuhu 241000 P. R. China

Chi-Wu Tang, Min Tang, and Lei Wu
Department of Mathematics
Anhui Normal University
Wuhu 241000
P. R. China