On the values of representation functions II

Abstract

For a set $A$ of nonnegative integers, let $R_2(A,n)$ and $R_3(A,n)$ denote the number of solutions to $n=a+a^{\prime }$ with $a,a^{\prime }\in A$, $a<a^{\prime }$ and $a\le a^{\prime }$, respectively. In this paper, we prove that, if $A\subseteq \mathbb{N}$ and $N$ is a positive integer such that $R_2(A,n)=R_2(\mathbb{N}\setminus A,n)$ for all $n\ge 2N-1$, then for any $\theta$ with $0<\theta <\frac{2\log 2-\log 3}{42\log 2-9\log 3}$, the set of integers $n$ with $R_2(A,n)=\frac{n}{8}+O(n^{1-\theta })$ has density one. The similar result holds for $R_3(A,n)$. These improve the results of the first author.

12 pages