Note on a result of Chen and Lev

Abstract

Let ${\mathbb {N}}$ be the set of all nonnegative integers. For a given set $S\subset {\mathbb {N}}$ the representation function $R_S(n)$ is defined as the number of solutions of the equation $n=s+s'$, $s<s'$, $s,s'\in S$. Let l be a positive integer. In this paper, we prove that there exist sets A and B such that ${\mathbb {N}}=A\cup B$ and $A\cap B=(2^{2l}-1)+(2^{2l+1}-1){\mathbb {N}}$ and $R_A(n)=R_B(n)\ge 1$ for every positive integer n except for finite exceptions. This result improves the result of Chen and Lev.