Théorie des nombres
Green’s problem on additive complements of the squares
Comptes Rendus. Mathématique, Tome 358 (2020) no. 8, pp. 897-900.

Let $A$ and $B$ be two subsets of the nonnegative integers. We call $A$ and $B$ additive complements if all sufficiently large integers $n$ can be written as $a+b$, where $a\in A$ and $b\in B$. Let $S=\left\{{1}^{2},{2}^{2},{3}^{2},···\right\}$ be the set of all square numbers. Ben Green was interested in the additive complement of $S$. He asked whether there is an additive complement $B={\left\{{b}_{n}\right\}}_{n=1}^{\infty }\subseteq ℕ$ which satisfies ${b}_{n}=\frac{{\pi }^{2}}{16}{n}^{2}+o\left({n}^{2}\right)$. Recently, Chen and Fang proved that if $B$ is such an additive complement, then

 $\underset{n\to \infty }{lim sup}\phantom{\rule{0.166667em}{0ex}}\frac{\frac{{\pi }^{2}}{16}{n}^{2}-{b}_{n}}{{n}^{1/2}logn}\ge \sqrt{\frac{2}{\pi }}\frac{1}{log4}.$

They further conjectured that

 $\underset{n\to \infty }{lim sup}\phantom{\rule{0.166667em}{0ex}}\frac{\frac{{\pi }^{2}}{16}{n}^{2}-{b}_{n}}{{n}^{1/2}logn}=+\infty .$

In this paper, we confirm this conjecture by giving a much more stronger result, i.e.,

 $\underset{n\to \infty }{lim sup}\phantom{\rule{0.166667em}{0ex}}\frac{\frac{{\pi }^{2}}{16}{n}^{2}-{b}_{n}}{n}\ge \frac{\pi }{4}.$

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DOI : https://doi.org/10.5802/crmath.107
Classification : 11B13,  11B75
@article{CRMATH_2020__358_8_897_0,
author = {Yuchen Ding},
title = {Green{\textquoteright}s problem on additive complements of the squares},
journal = {Comptes Rendus. Math\'ematique},
pages = {897--900},
publisher = {Acad\'emie des sciences, Paris},
volume = {358},
number = {8},
year = {2020},
doi = {10.5802/crmath.107},
language = {en},
}
Yuchen Ding. Green’s problem on additive complements of the squares. Comptes Rendus. Mathématique, Tome 358 (2020) no. 8, pp. 897-900. doi : 10.5802/crmath.107. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.5802/crmath.107/

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