On the structure of sets in a residue class ring with the same representation function

Abstract

Let m be an integer with m≥2. For a given set S⊆Zm and n‾∈Zm, let RS(n‾) denote the number of solutions of the equation n‾=s‾+s′‾ with unordered pair (s‾,s′‾)∈S×S and s‾≠s′‾. In this paper, we determine the structure of sets A,B satisfying A∪B=Zm, |A∩B|=2 and RA(n‾)=RB(n‾) for all n‾∈Zm, where m is even.