Abstract:

For $n\in \mathbb{Z}, A\subset \mathbb{Z}$, let $\delta_{A}(n)$ denote the number of representations of $n$ in the form $n=a-a'$, 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 $(\mathbb{Z},-)$.