ON A PROBLEM OF CHEN AND LEV

Abstract

For a given set$S\subset \mathbb{N}$,$R_{S}(n)$is the number of solutions of the equation$n=s+s^{\prime },s<s^{\prime },s,s^{\prime }\in S$. Suppose that$m$and$r$are integers with$m>r\geq 0$and that$A$and$B$are sets with$A\cup B=\mathbb{N}$and$A\cap B=\{r+mk:k\in \mathbb{N}\}$. We prove that if$R_{A}(n)=R_{B}(n)$for all positive integers$n$, then there exists an integer$l\geq 1$such that$r=2^{2l}-1$and$m=2^{2l+1}-1$. This solves a problem of Chen and Lev [‘Integer sets with identical representation functions’,Integers16(2016), A36] under the condition$m>r$.