论文标题
关于$ \ mathbb {z} _ {m} $的分区问题，具有相同的表示功能
On a problem of partitions of $\mathbb{Z}_{m}$ with the same representation functions
论文作者
论文摘要
对于任何积极的整数$ m $，令$ \ mathbb {z} _ {m} $是一组残基类Modulo $ m $。对于$ a \ subseteq \ mathbb {z} _ {m} $和$ \ overline {n} \ in \ mathbb {z} _ {m {m} $，让表示函数$ r_ {a} $ \ OVERLINE {n} = \ OVERLINE {A}+\ edline {a'} $带有序对$（\ overline {a}，\ overline {a'}）\ in A \ times a $。在本文中，我们确定所有集合$ a，b \ subseteq \ mathbb {z} _ {m} $ a \ a \ cup b = \ mathbb {z} _ {m} _ {m} $和$ | a \ cap b | = 2 $或$ m-2 $ $ r_ {a}（\ overline {n}）= r_ {b}（\ overline {n}）$ for All $ \ Overline {n} \ in \ Mathbb {z} _ {m {m} $。我们还证明，如果$ m $是一个积极的整数，$ 4 | m $，那么存在两个不同的集合$ a，b \ subseteq \ mathbb {z} _ {m} _ {m} $，带有$ a \ cup b = \ cup b = \ mathbb {z} _ {z} _ {m} _ {m} $和$ | a \ cap b | a+\ edline {\ frac {m} {2}} $，使得$ r_ {a}（\ overline {n}）= r_ {b}（\ edlline {n}）$ $ \ edimelline {n} $如果$ m $是一个正整数，$ 2 \ | m $，$ a \ cup b = \ mathbb {z} _ {m} $和$ | a \ cap b | = 4 $或$ m-4 $，则$ r_ {a} \ Mathbb {z} _ {m} $，仅当$ b = a+\ overline {\ frac {\ frac {m} {2}} $。
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 $B=A+\overline{\frac{m}{2}}$.
