The lower bound of weighted representation function

For any given set A of nonnegative integers and for any given two positive integers \(k_1,k_2\) $k_1,k_2$ , \(R_{k_1,k_2}(A,n)\) $R_{k_1,k_2}(A,n)$ is defined as the number of solutions of the equation \(n=k_1a_1+k_2a_2\) $n=k_1{a}_1+k_2{a}_2$ with \(a_1,a_2\in A\) $a_1,a_2\in A$ . In this paper, we prove that if integer \(k\ge 2\) $k\ge 2$ and set \(A\subseteq {\mathbb {N}}\) $A\subseteq \mathbb{N}$ such that \(R_{1,k}(A,n)=R_{1,k}({\mathbb {N}}\setminus A,n)\) $R_{1,k}(A,n)=R_{1,k}(\mathbb{N}\backslash A,n)$ holds for all integers \(n\ge n_0\) $n\ge n_0$ , then \(R_{1,k}(A,n)\gg \log n\) $R_{1,k}(A,n)\gg logn$ .