Journal of Integer Sequences, Vol. 14 (2011), Article 11.1.8

Unique Difference Bases of Z

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


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},-)$.

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

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