On a problem of partitions of $\mathbb{Z}_{m}$ with the same representation functions

TL;DR

This paper characterizes pairs of subsets of residue classes modulo m that produce identical representation functions, focusing on specific intersection sizes and the structure of such sets.

Contribution

It determines all such pairs with given intersection sizes and explores conditions for their equivalence, especially when m is divisible by 4 or has a specific prime factorization.

Findings

Characterization of sets with equal representation functions for intersection sizes 2 and m-2.

Existence of distinct sets with equal representation functions when m is divisible by 4.

Condition for equality of representation functions when m is even, involving a shift by m/2.

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 representation function $R_{A}(\overline{n})$ denote the number of solutions of the equation $\overline{n}=\overline{a}+\overline{a'}$ with ordered pairs $(\overline{a}, \overline{a'})\in A \times A$. In this paper, we determine all sets $A, B\subseteq \mathbb{Z}_{m}$ with $A\cup B=\mathbb{Z}_{m}$ and $|A\cap B|=2$ or $m-2$ such that $R_{A}(\overline{n})=R_{B}(\overline{n})$ for all $\overline{n}\in \mathbb{Z}_{m}$. We also prove that if $m$ is a positive integer with $4|m$, then there exist two distinct sets $A, B\subseteq \mathbb{Z}_{m}$ with $A\cup B=\mathbb{Z}_{m}$ and $|A\cap B|=4$ or $m-4$, $B\neq A+\overline{\frac{m}{2}}$ such that $R_{A}(\overline{n})=R_{B}(\overline{n})$ for all $\overline{n}\in \mathbb{Z}_{m}$. If $m$ is a positive integer with $2\|m$, $A\cup B=\mathbb{Z}_{m}$ and $|A\cap B|=4$ or $m-4$, then $R_{A}(\overline{n})=R_{B}(\overline{n})$ for all $\overline{n}\in \mathbb{Z}_{m}$ if and only if $B=A+\overline{\frac{m}{2}}$.