A note on additive complements of the squares

Let $\mathcal{S}=\{1^2,2^2,3^2,\mathrm{.}\mathrm{.}\mathrm{.}\}$ be the set of squares and $\mathcal{W}=\{w_n{\}}_{n=1}^{\mathrm{\infty }}\subset \mathbb{N}$ be an additive complement of $\mathcal{S}$ so that $\mathcal{S}+\mathcal{W}\supset \{n\in \mathbb{N}:n\ge N_0\}$ for some $N_0$. Let ${\mathcal{R}}_{\mathcal{S},\mathcal{W}}(n)=\mathrm{\#}\{(s,w):n=s+w,s\in \mathcal{S},w\in \mathcal{W}\}$. In 2017, Chen-Fang \cite{C-F} studied the lower bound of ${\sum }_{n=1}^N{R}_{\mathcal{S},\mathcal{W}}(n)$. In this note, we improve Cheng-Fang’s result and get that $$\sum _{n=1}^N{R}_{\mathcal{S},\mathcal{W}}(n)-N\gg N^{1\mathrm{/}2}\mathrm{.}$$ As an application, we make some progress on a problem of Ben Green problem by showing that $$\underset{n\to \mathrm{\infty }}{\mathrm{lim\hspace{0.17em}sup}}\frac{\frac{{\pi }^2}{16}{n}^2-w_n}{n}\ge \frac{\pi }{4}+\frac{0.193{\pi }^2}{8}\mathrm{.}$$