A problem of Yang and Chen on weighted representation functions

Abstract

Let $\mathbb{N}$ denote the set of nonnegative integers. For an integer $k>1$ and a set $A\subseteq \mathbb{N}$, let $R_{1,k}(A,n)$ denote the number of solutions of $n=a_1+k{a}_2$ with $a_1,a_2\in A$. For integers $k>1$ and $t\ge 1$, Yang and Chen defined $f_k(t)$ to be the number of sets $A\subseteq \mathbb{N}$ for which \[ R_{1,k}(A,n)=R_{1,k}(\mathbb N\setminus A,n) \] for all integers $n\ge t$, and asked whether $f_k(t)$ and $f_l(t)$ are eventually equal for any integers $k,l>1$. We prove that \[ f_k(t)\asymp_k \frac{2^t}{t^{k/2}}, \] which gives a negative answer to their problem.