Constructing finite sets from their representation functions

Abstract

For any positive integer m, let $\mathbb{Z}_{m}$ be the set of residue classes modulo m. For $A\subseteq \mathbb{Z}_{m}$ and $\overline{n}\in \mathbb{Z}_{m}$, let the representation function $R_{A}(\overline{n})$ denote the number of solutions of the equation $\overline{n}=\overline{a}+\overline{a'}$ with unordered pairs $(\overline{a}, \overline{a'})\in A \times A$. We characterize the partitions of $\mathbb{Z}_{2p}$ with $A\cup B=\mathbb{Z}_{2p}$ and $|A\cap B|=2$ such that $R_{A}(\overline{n})=R_{B}(\overline{n})$ for all $\overline{n}\in\mathbb{Z}_{2p}$, where p is an odd prime.