On the values of representation functions

Abstract

For a set A of nonnegative integers, let $R_2(A,n)$ and $R_3(A,n)$ denote the number of solutions to $n=a+a'$ with $a,a'\in A$, ${a< a'}$ and ${a,a'\in A}$, ${a\le a'}$, respectively. Recently, some mathematicians devoted themselves to research of $R_2(A,n)$ and $R_3(A,n)$ and obtained many significant results. 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 $\varepsilon >0$, the set of integers n with $(\frac{1}{8}-\varepsilon )n\le R_2(A,n)\le (\frac{1}{8}+\varepsilon )n$ has density one. A similar result is true for $R_3(A,n)$.