×

Reconstructing integer sets from their representation functions. (English) Zbl 1068.11006

First proofs of two theorems of G. Dombi [Acta Arith. 103, No. 2, 137–146 (2002; Zbl 1014.11009)] and Y.-G. Chen and 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 \(i,j,z\in\mathbb{N}\).

MSC:

11B34 Representation functions
05A17 Combinatorial aspects of partitions of integers
11B13 Additive bases, including sumsets
PDFBibTeX XMLCite
Full Text: EuDML EMIS