On the values of representation functions II

For a set A of nonnegative integers, let R2(A,n) and R3(A,n) denote the number solutions to n=a+a′ with a,a′∈A, a<a′ a≤a′, respectively. In this paper, we prove that, if A⊆N N is positive integer such that R2(A,n)=R2(N∖A,n) for all n≥2N−1, then any θ 0<θ<2log⁡2−log⁡342log⁡2−9log⁡3, cardinality n≤x R2(A,n)=n8+O(n1−θ) equals x+O(x1−δ) some constant δ. The similar result holds R3(A,n). These improve results first author.