Green’s problem on additive complements of the squares

Ding, Yuchen

doi:10.5802/crmath.107

Théorie des nombres

Green’s problem on additive complements of the squares

Ding, Yuchen ¹

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 $S = {1^{2}, 2^{2}, 3^{2}, \cdot \cdot \cdot}$ 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 $B = {b_{n}}_{n = 1}^{\infty} \subseteq ℕ$ which satisfies $b_{n} = \frac{π^{2}}{16} n^{2} + o (n^{2})$ . Recently, Chen and Fang proved that if $B$ is such an additive complement, then

\underset{n \to \infty}{lim sup} \frac{\frac{π^{2}}{16} n^{2} - b_{n}}{n^{1 / 2} log n} \geq \sqrt{\frac{2}{π}} \frac{1}{log 4} .

They further conjectured that

\underset{n \to \infty}{lim sup} \frac{\frac{π^{2}}{16} n^{2} - b_{n}}{n^{1 / 2} log n} = + \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{π^{2}}{16} n^{2} - b_{n}}{n} \geq \frac{π}{4} .

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

Affiliations des auteurs :

Ding, Yuchen ¹

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

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     title = {Green{\textquoteright}s problem on additive complements of the squares},
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Ding, Yuchen. 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. http://www.numdam.org/articles/10.5802/crmath.107/

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