The lower bound of weighted representation function

Abstract

For any given set A of nonnegative integers and for any given two positive integers $k_1,k_2$, $R_{k_1,k_2}(A,n)$ is defined as the number of solutions of the equation $n=k_1a_1+k_2a_2$ with $a_1,a_2\in A$. In this paper, we prove that if integer $k\ge 2$ and set $A\subseteq {\mathbb {N}}$ such that $R_{1,k}(A,n)=R_{1,k}({\mathbb {N}}\setminus A,n)$ holds for all integers $n\ge n_0$, then $R_{1,k}(A,n)\gg \log n$.