Some Properties Of Sets With Same Representation Functions

Let N denote the set of nonnegative integers.Let A be a subset of N and n a nonnegative integer.The representation function R_A?n?is the number of solutions of the equation n=a₁+a₂,where a₁,a₂?A,a₁<a₂.Chen and Lev constructed two sets A and B,which have identical representation functions,satisfying that A?B=N and the intersection of A and B is an infinite arithmetic progression.In this paper,we continue to study the properties of the two sets.We prove that x can always be the sum of two different integers in A if x is sufficiently large,and prove that R_A?x?approaches to infinity when x approaches to infinity.