On the Structure of Finite Sets with Identical Representation Functions

For any positive integer m, let ℤm be the set of residue classes modulo m. For A ⊆ ℤm and n̄ₘ , let the representation function RA(n̄) denote the number of solutions of the equation n̄=ā+a^ with unordered pairs (ā,a^)∈ A× A . For any integer a with (a, m) = 1, let ordm(a) be the least positive integer h such that ah ≡ 1 (mod m). Let m = 2αM, 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 ∣ ordp (2) for some odd prime p with p ≥ M, then there exist two distinct sets A, B ⊆ ℤm with A ∪ B = ℤm, ∣A ∩ B∣ = 2 and B≠ A+2α-1M such that RA(n̄)=RB(n̄) for all n̄=Zₘ . We also prove that if 2 ∤ ordp(2) for any odd prime p with p ∣ M and A, B ⊆ ℤm with A ∪ B = ℤm, ∣A ∩ B∣ =2, then RA(n̄)=RB(n̄) for all n̄∈ Zₘ if and only if B = A+2α-1M .