A problem of Yang and Chen on weighted representation functions

Source 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 $n=a_1+ka_2$ with $a_1,a_2\in A$. For integers $k>1$ and $t\ge1$, 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.