论文标题
关于正方形的添加补充的注释
A note on additive complements of the squares
论文作者
论文摘要
令$ \ MATHCAL {S} = \ {1^2,2^2,3^2，... \} $是一组平方，$ \ Mathcal {w} = \ {w_n \} _ {n = 1}因此，对于某些$ n_0 $，$ \ MATHCAL {S} + \ MATHCAL {W} \ SUPSET \ {N \ in \ MATHBB {N}：n \ geq n_0 \} $对于某些$ n_0 $。令$ \ Mathcal {r} _ {\ Mathcal {s}，\ Mathcal {w}}（n）= \＃\ {（s，s，w）：n = s+w，s+w，s \ in \ mathcal {s}，s s}，w \ in \ mathcal {w \ in \ mathcal {w} {w} \} $。 在2017年，Chen-fang \ cite {c-f}研究了$ \ sum_ {n = 1}^nr _ {\ Mathcal {s}，\ Mathcal {w}}}（n）$的下限。在本说明中，我们改善了Cheng-fang的结果，并获得$ \ sum_ {n = 1}^nr _ {\ Mathcal {s}，\ Mathcal {w}}（w}}（n）-n \ gg n \ gg n^{1/2}。 $ \ limsup_ {n \ rightarrow \ infty} \ frac {\ frac {π^2} {16} n^2-w_n} {n} {n} {n} \ ge \ ge \fracπ{4}+\ frac+\ frac {0.193π^2} {0.193π^2} {8} {8} {8}
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 $$\limsup_{n\rightarrow\infty}\frac{\frac{π^2}{16}n^2-w_n}{n}\ge \fracπ{4}+\frac{0.193π^2}{8}.$$
