Reconstructing Integer Sets From Their Representation Functions

Abstract

We give a simple common proof to recent results by Dombi and by Chen and Wang concerning the number of representations of an integer in the form $a_1+a_2$, where $a_1$ and $a_2$ are elements of a given infinite set of integers. Considering the similar problem for differences, we show that there exists a partition ${\Bbb N}=\cup_{k=1}^\infty A_k$ of the set of positive integers such that each $A_k$ is a perfect difference set (meaning that any non-zero integer has a unique representation as $a_1-a_2$ with $a_1,a_2\in A_k$). A number of open problems are presented.