Abstract

For n ∈ ℤ, A ⊂ ℤ, let δ_{A}(n) denote the number of representations of n in the form n = a − a′, where a, a′ ∈ A. A set A ⊂ ℤ is called a unique difference basis of ℤ if δ_{A}(n) = 1 for all n ≠ 0 in ℤ. In this paper, we prove that there exists a unique difference basis of ℤ whose growth is logarithmic. These results show that the analogue of the Erdős-Turán conjecture fails to hold in (ℤ, −).