Reconstructing integer sets from their representation functions (Q1773171)

First proofs of two theorems of \textit{G. Dombi} [Acta Arith. 103, No. 2, 137--146 (2002; Zbl 1014.11009)] and \textit{Y.-G. Chen} and \textit{B. Wang} [Acta Arith. 110, No. 3, 299--303 (2003; Zbl 1032.11008)] are given. Then the parallel problem for differences is investigated. A set \(A\subseteq\mathbb{Z}\) is a perfect difference set if any integer \(\neq 0\) has a unique representation as a difference of two elements of \(A\). Then the following theorem is proved: There is a partition \(\mathbb{N}=\bigcup^\infty_{k=1} A_k\) such that each \(A_k\) is a perfect difference set and \(|A_i\cap(A_j+ z)|\leq 2\) for any \(i,j,z\in\mathbb{N}\).