Comptes Rendus
no. 8
Théorie des nombres
Green’s problem on additive complements of the squares
Yuchen Ding1
1 School of Mathematical Science, Yangzhou University, Yangzhou 225002, P. R. China
Comptes Rendus. Mathématique, Volume 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∈A and b∈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={b n } n=1 ∞ ⊆ℕ which satisfies b n =π 2 16n 2 +o(n 2 ). Recently, Chen and Fang proved that if B is such an additive complement, then

lim sup n→∞ π 2 16n 2 -b n n 1/2 logn≥2 π1 log4.

They further conjectured that

lim sup n→∞ π 2 16n 2 -b n n 1/2 logn=+∞.

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

lim sup n→∞ π 2 16n 2 -b n n≥π 4.

Reçu le :
Révisé le :
Accepté le :
Publié le :
DOI : 10.5802/crmath.107
Classification : 11B13, 11B75
Yuchen Ding 1

1 School of Mathematical Science, Yangzhou University, Yangzhou 225002, P. R. China
Licence : CC-BY 4.0
Droits d'auteur : Les auteurs conservent leurs droits 
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Yuchen Ding. Green’s problem on additive complements of the squares. Comptes Rendus. Mathématique, Volume 358 (2020) no. 8, pp. 897-900. doi : 10.5802/crmath.107. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.5802/crmath.107/

[1] Ramachandran Balasubramanian On the additive completion of squares, J. Number Theory, Volume 29 (1988) no. 1, pp. 10-12 | DOI | MR | Zbl

[2] Ramachandran Balasubramanian; D. S. Ramana Additive complements of the squares, C. R. Math. Acad. Sci., Soc. R. Can., Volume 23 (2001) no. 1, pp. 6-11 | MR | Zbl

[3] Ramachandran Balasubramanian; Kannan Soundararajan On the additive completion of squares. II., J. Number Theory, Volume 40 (1992) no. 2, pp. 127-129 | DOI | MR | Zbl

[4] Yong-Gao Chen; Jin-Hui Fang Additive complements of the squares, J. Number Theory, Volume 180 (2017), pp. 410-422 | DOI | MR | Zbl

[5] Javier Cilleruelo The additive completion of k th powers, J. Number Theory, Volume 44 (1993) no. 3, pp. 237-243 | DOI | MR | Zbl

[6] Pál Erdős Problems and Results in Additive Number Theory, Colloque sur la Thérie des Nombres, Bruxelles, 1955, George Thone; Masson, 1956, pp. 127-137 | Zbl

[7] Laurent Habsieger On the additive completion of polynomial sets, J. Number Theory, Volume 51 (1995) no. 1, pp. 130-135 | DOI | MR | Zbl

[8] Leo Moser On the additive completion of sets of integers, Proceedings of Symposia in Pure Mathematics, Volume 8 (1965), pp. 175-180 | DOI | MR | Zbl

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