On the Structure of Finite Sets with Identical Representation Functions

For any positive integer m, let ℤ_m be the set of residue classes modulo m. For A ⊆ ℤ_m and \(\overline{n}\in\mathbb{Z}_{m}\) $$\overline{n}\in {\mathbb{Z}}_m$$ , let the representation function \(R_{A}(\overline{n})\) $$R_A(\overline{n})$$ denote the number of solutions of the equation \(\overline{n}=\overline{a}+\overline{a^{\prime}}\) $$\overline{n}=\overline{a}+\overline{a^{\mathrm{\prime }}}$$ with unordered pairs \((\overline{a},\overline{a^{\prime}})\in A\times A \) $$(\overline{a},\overline{a^{\mathrm{\prime }}})\in A\times A$$ . For any integer a with (a, m) = 1, let ord_m(a) be the least positive integer h such that a^h ≡ 1 (mod m). Let m = 2^αM, where α is an integer with α ≥ 2 and M is an odd integer with M ≥ 3. In this paper, we prove that if 2 ∣ ord_p (2) for some odd prime p with p ≥ M, then there exist two distinct sets A, B ⊆ ℤ_m with A ∪ B = ℤ_m, ∣A ∩ B∣ = 2 and \(B\ne A+\overline{2^{\alpha-1}M}\) $$B\ne A+\overline{2^{\alpha -1}M}$$ such that \(R_{A}(\overline{n})=R_{B}(\overline{n})\) $$R_A(\overline{n})=R_B(\overline{n})$$ for all \(\overline{n}=\mathbb{Z}_{m}\) $$\overline{n}={\mathbb{Z}}_m$$ . We also prove that if 2 ∤ ord_p(2) for any odd prime p with p ∣ M and A, B ⊆ ℤ_m with A ∪ B = ℤ_m, ∣A ∩ B∣ =2, then \(R_{A}(\overline{n})=R_{B}(\overline{n})\) $$R_A(\overline{n})=R_B(\overline{n})$$ for all \(\overline{n}\in \mathbb{Z}_{m}\) $$\overline{n}\in {\mathbb{Z}}_m$$ if and only if \(B = A+\overline{2^{\alpha-1}M}\) $$B=A+\overline{2^{\alpha -1}M}$$ .