On partitions of ${\mathbb{Z}}_m$ with the same representation function

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 $R_A(\overline{n})$ denote the number of solutions of $\overline{n}=\overline{a}+\overline{a^{\prime }}$ with unordered pairs $(\overline{a},\overline{a^{\prime }})\in A\times A$. In this paper, we prove that if $m=2^{\alpha }$ with $\alpha \ne 2$, $A\cup B={\mathbb{Z}}_m$ and $|A\cap B|=2$, then $R_A(\overline{n})=R_A(\overline{n})$ for all $\overline{n}\in {\mathbb{Z}}_m$ if and only if $B=A+\overline{\frac{m}{2}}$.

9 pages