Residue Class Ring with Identical Representation Function

Abstract

For an integer m ≥ 2, let ℤ/mℤ be the set of all residue classes mod m. For S ⊆ ℤ/mℤ and $\bar{n}\in\mathbb{Z}/m\mathbb{Z},\,R_{S}(\bar{n})$ is defined as the number of solutions to the equation $\bar{n}=\bar{s}+\bar{s^{\prime}}$ with an unordered pair $(\bar{s},\bar{s^{\prime}})\in S^{2}$ and $\bar{s}\ne\bar{s^{\prime}}$. In this paper, the author determines the structures of sets A and B such that A ⋃ B = ℤ/mℤ, $A\;\cap\;B=\bar{k}\mathbb{Z}$ and $R_{A}(\bar{n})=R_{B}(\bar{n})$ for all $\bar{n}\in\mathbb{Z}/m\mathbb{Z}$, where k is an integer.

Keywords

Residue class / Representation function / 11B13

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