Abstract For a given set S ⊆ Z m and n ¯ ∈ Z m , R S ( n ¯ ) is defined as the number of solutions of the equation n ¯ = s ¯ + s ′ ¯ with unordered pair ( s ¯ , s ′ ¯ ) ∈ S 2 and s ¯ ≠ s ′ ¯ . In this paper, we prove that if m is an even positive integer and not a power of 2 then there exist two distinct sets A , B with A ∪ B = Z m , | A ∩ B | = 2 , B ≠ A + m ∕ 2 ¯ such that R A ( n ¯ ) = R B ( n ¯ ) for all n ¯ ∈ Z m . If m is a power of 2, A ∪ B = Z m , | A ∩ B | = 2 , then R A ( n ¯ ) = R B ( n ¯ ) for all n ¯ ∈ Z m if and only if B = A + m ∕ 2 ¯ .

中文翻译：

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关于具有相同表示函数的 Zm 分区的某些性质

摘要 对于给定的集合 S ⊆ Z m 和 n ¯ ∈ Z m ，RS ( n ¯ ) 被定义为方程 n ¯ = s ¯ + s ′ ¯ 的解数，其中有无序对 ( s ¯ , s ′ ¯ ) ∈ S 2 且 s¯ ≠ s ′¯ 。在本文中，我们证明如果 m 是偶数正整数而不是 2 的幂，那么存在两个不同的集合 A , B 且 A ∪ B = Z m , | A∩B | = 2 , B ≠ A + m ∕ 2 ¯ 使得 RA ( n ¯ ) = RB ( n ¯ ) 对于所有 n ¯ ∈ Z m 。如果 m 是 2 的幂，则 A ∪ B = Z m ，| A∩B | = 2 ，则 RA ( n ¯ ) = RB ( n ¯ ) 对于所有 n ¯ ∈ Z m 当且仅当 B = A + m ∕ 2 ¯ 。