A note on the lower bound of representation functions

For a set $A$ of nonnegative integers, let $R_2(A,n)$ denote the number of solutions to $n=a+a^{\mathrm{\prime }}$ with $a,a^{\mathrm{\prime }}\in A$, $a<a^{\mathrm{\prime }}$. Let $A_0$ be the Thue-Morse sequence and $B_0=\mathbb{N}\setminus A_0$. Let $A\subset \mathbb{N}$ and $N$ be a positive integer such that $R_2(A,n)=R_2(\mathbb{N}\setminus A,n)$ for all $n\ge 2N-1$. Previously, the first author proved that if $\mathrm{\mid }A\cap A_0\mathrm{\mid }=+\mathrm{\infty }$ and $\mathrm{\mid }A\cap B_0\mathrm{\mid }=+\mathrm{\infty }$, then $R_2(A,n)\ge \frac{n+3}{56N-52}-1$ for all $n\ge 1$. In this paper, we prove that the above lower bound is nearly best possible. We also get some other results.