For
![$ n\in \mathbb{Z}, A\subset \mathbb{Z}$](abs/img1.gif)
, let
![$ \delta_{A}(n)$](abs/img2.gif)
denote the number of representations of
![$ n$](abs/img3.gif)
in the form
![$ n=a-a'$](abs/img4.gif)
,
where
![$ a,a'\in A$](abs/img5.gif)
. A set
![$ A\subset \mathbb{Z}$](abs/img6.gif)
is called a unique
difference basis of
![$ \mathbb{Z}$](abs/img7.gif)
if
![$ \delta_{A}(n)=1$](abs/img8.gif)
for all
![$ n\neq
0$](abs/img9.gif)
in
![$ \mathbb{Z}$](abs/img7.gif)
. In this paper, we prove that there exists a unique
difference basis of
![$ \mathbb{Z}$](abs/img7.gif)
whose growth is logarithmic. These
results show that the analogue of the Erdos-Turán conjecture
fails to hold in
![$ (\mathbb{Z},-)$](abs/img10.gif)
.
Received October 12 2010;
revised version received January 26 2011.
Published in Journal of Integer Sequences, February 9 2011.