On the structure of finite sets with the same representation functions.

For any positive integer m , let Z m be the set of residue classes modulo m. For A ⊆ Z m and n ‾ ∈ Z m , let the representation function R A (n ‾) denote the number of solutions of the equation n ‾ = a ‾ + a ′ ‾ with unordered pairs (a ‾ , a ′ ‾) ∈ A × A. Let M be an odd integer with M ≥ 3 and m = 2 M. In this paper, we determine the structure of A , B ⊆ Z m with A ∪ B = Z m and | A ∩ B | = 2 such that R A (n ‾) = R B (n ‾) for all n ‾ ∈ Z m. [ABSTRACT FROM AUTHOR]