A note on additive complements of the squares

Abstract

Let S={12,22,32,...}\mathcal{S}=\{1^2,2^2,3^2,...\} be the set of squares and W={wn}n=1βˆžβŠ‚N\mathcal{W}=\{w_n\}_{n=1}^{\infty} \subset \mathbb{N} be an additive complement of S\mathcal{S} so that S+WβŠƒ{n∈N:nβ‰₯N0}\mathcal{S} + \mathcal{W} \supset \{n \in \mathbb{N}: n \geq N_0\} for some N0N_0. Let RS,W(n)=#{(s,w):n=s+w,s∈S,w∈W}\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 βˆ‘n=1NRS,W(n)\sum_{n=1}^NR_{\mathcal{S},\mathcal{W}}(n). In this note, we improve Cheng-Fang's result and get that βˆ‘n=1NRS,W(n)βˆ’N≫N1/2.\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 lim sup⁑nβ†’βˆžΟ€216n2βˆ’wnnβ‰₯Ο€4+0.193Ο€28.\limsup_{n\rightarrow\infty}\frac{\frac{\pi^2}{16}n^2-w_n}{n}\ge \frac{\pi}{4}+\frac{0.193\pi^2}{8}.Comment: The new version significantly improves the result of the former on

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