论文标题
在$ \ mathbb {z} _ {m} $的分区中，具有相同的表示功能
On partitions of $\mathbb{Z}_{m}$ with the same representation function
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} $ $ \ OVERLINE {n} = \ OVERLINE {a}+\ edline {a'} $带有未排序的Pairs $（\ overline {a}，\ overline {a'}）\ in A \ times a $。在本文中，我们证明，如果$ m = 2^α$带有$α\ neq 2 $，$ a \ cup b = \ mathbb {z} _ {m} $和$ | a \ cap b | = 2 $，然后$ r_ {a} $ \ OVERLINE {n} \ in \ MATHBB {z} _ {m} $ if，并且仅当$ b = a+\ overline {\ 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 $R_{A}(\overline{n})$ 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 $m=2^α$ with $α\neq 2$, $A\cup B=\mathbb{Z}_{m}$ 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 $B=A+\overline{\frac{m}{2}}$.
