On certain properties of partitions of $\mathbb {Z}_m$ with the same representation function, II

Abstract

For a given set $S\subseteq \mathbb {Z}_m$ and $\overline{n}\in \mathbb {Z}_m$, $R_S(\overline{n})$ is defined as the number of solutions of the equation $\overline{n}=\overline{s}+\overline{s'}$ with unordered pair $(\overline{s},\overline{s'})\in S^2$ and $\overline{s}\ne \overline{s'}$. In this paper, we prove that if $m=2^id,\,i\ge 1,\,2\not \mid d, d > 1$ then there exist two sets $A,B\subseteq \mathbb {Z}_{m}$ with $A\cup B=\mathbb {Z}_{m}$, $A\cap B=\{\overline{r_1}<\overline{r_2}\}$, $B\ne A+\overline{m/2}$ and $R_{A}(\overline{n})=R_{B}(\overline{n})$ for all $\overline{n}\in \mathbb {Z}_{m}$ if and only if there exists an odd integer $u\ne d,~0<u<2d$ such that $\overline{r_2}-\overline{r_1}=\overline{2^{i-1}u}$.