A note on additive complements of the squares

Abstract: Let $\mathcal{S}={1^2,2^2,3^2,...}$ be the set of squares and $\mathcal{W}={w_n}{n=1}^{\infty} \subset \mathbb{N}$ be an additive complement of $\mathcal{S}$ so that $\mathcal{S} + \mathcal{W} \supset {n \in \mathbb{N}: n \geq N_0}$ for some $N_0$. Let $\mathcal{R}{\mathcal{S},\mathcal{W}}(n) = #{(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 $\sum_{n=1}^NR_{\mathcal{S},\mathcal{W}}(n)$. In this note, we improve Cheng-Fang's result and get that $$\sum_{n=1}^NR_{\mathcal{S},\mathcal{W}}(n)-N\gg N^{1/2}.$$ As an application, we make some progress on a problem of Ben Green problem by showing that $$\limsup_{n\rightarrow\infty}\frac{\frac{\pi^2}{16}n^2-w_n}{n}\ge \frac{\pi}{4}+\frac{0.193\pi^2}{8}.$$