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 $$\limsup_{ n\rightarrow\infty}\frac{ \frac{ \pi^2}{ 16}n^2-w_n}{ n}\ge \frac{ \pi}{ 4}+\frac{ 0.193\pi^2}{ 8}.$$
Categories: math.NT