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

TL;DR

This paper characterizes when two partitions of the residue class set modulo a power of two have identical representation functions, showing a specific shift relationship between the sets under certain conditions.

Contribution

It provides a precise characterization of partitions with equal representation functions for powers of two, extending understanding of additive properties in modular arithmetic.

Findings

If $m=2^{eta}$ with $eta eq 2$, then $A$ and $B$ have equal representation functions iff $B=A+rac{m}{2}$.

The result holds for all residue classes when the sets differ by a specific shift.

The case $m=4$ is excluded from the main theorem.

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 $\overline{n}=\overline{a}+\overline{a'}$ with unordered pairs $(\overline{a}, \overline{a'})\in A \times A$. In this paper, we prove that if $m=2^{\alpha}$ with $\alpha\neq 2$, $A\cup B=\mathbb{Z}_{m}$ 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 $B=A+\overline{\frac{m}{2}}$.