Théorie des nombres

Green’s problem on additive complements of the squares

Ding, Yuchen¹

¹ School of Mathematical Science, Yangzhou University, Yangzhou 225002, P. R. China

Comptes Rendus. Mathématique, Tome 358 (2020) no. 8, pp. 897-900.

Résumé

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 $S=\{1^2,2^2,3^2,···\}$ 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 $B={\left\{b_n\right\}}_{n=1}^{\infty }\subseteq \mathbb{N}$ 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

$$\underset{n\to \infty }{lim\; sup}\frac{\frac{{\pi }^2}{16}{n}^2-b_n}{n^{1/2}logn}\ge \sqrt{\frac{2}{\pi }}\frac{1}{log4}.$$

They further conjectured that

$$\underset{n\to \infty }{lim\; sup}\frac{\frac{{\pi }^2}{16}{n}^2-b_n}{n^{1/2}logn}=+\infty .$$

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

$$\underset{n\to \infty }{lim\; sup}\frac{\frac{{\pi }^2}{16}{n}^2-b_n}{n}\ge \frac{\pi }{4}.$$

Détail

Reçu le : 2020-08-03
Révisé le : 2020-08-19
Accepté le : 2020-08-20
Publié le : 2020-12-03

DOI : 10.5802/crmath.107

Classification : 11B13, 11B75

Licence : CC-BY 4.0

Droits d'auteur : Les auteurs conservent leurs droits

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