The lower bound of weighted representation function

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 \(R_{1,k}(A,n)\gg \log n\).