Geodesic
Unique difference bases of \(\mathbb Z\)
Journal of integer sequences, Tome 14 (2011) no. 1
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},-)$.
Keywords: erdacute$\Acute $os-turán conjecture, difference bases, counting function 
@article{JIS_2011__14_1_a4,
     author = {Tang,  Chi-Wu and Tang,  Min and Wu,  Lei},
     title = {Unique difference bases of \(\mathbb {Z\)}},
     journal = {Journal of integer sequences},
     year = {2011},
     volume = {14},
     number = {1},
     zbl = {1223.11015},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/JIS_2011__14_1_a4/}
}
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Tang,  Chi-Wu; Tang,  Min; Wu,  Lei. Unique difference bases of \(\mathbb Z\). Journal of integer sequences, Tome 14 (2011) no. 1. http://geodesic.mathdoc.fr/item/JIS_2011__14_1_a4/