logo CRAS
Comptes Rendus. Mathématique

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 aA and bB. 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 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 supnπ216n2-bnn1/2logn2π1log4.

They further conjectured that

lim supnπ216n2-bnn1/2logn=+.

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

lim supnπ216n2-bnnπ4.

Reçu le :
Révisé le :
Accepté le :
Publié le :
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/

[1] Ramachandran Balasubramanian On the additive completion of squares, J. Number Theory, Volume 29 (1988) no. 1, pp. 10-12 | Article | MR 938866 | Zbl 0645.10047

[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 1816458 | Zbl 1007.11005

[3] Ramachandran Balasubramanian; Kannan Soundararajan On the additive completion of squares. II., J. Number Theory, Volume 40 (1992) no. 2, pp. 127-129 | Article | MR 1149732 | Zbl 0749.11021

[4] Yong-Gao Chen; Jin-Hui Fang Additive complements of the squares, J. Number Theory, Volume 180 (2017), pp. 410-422 | Article | MR 3679805 | Zbl 1421.11014

[5] Javier Cilleruelo The additive completion of k th powers, J. Number Theory, Volume 44 (1993) no. 3, pp. 237-243 | Article | MR 1233285 | Zbl 0786.11008

[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 0073.03102

[7] Laurent Habsieger On the additive completion of polynomial sets, J. Number Theory, Volume 51 (1995) no. 1, pp. 130-135 | Article | MR 1321728 | Zbl 0818.11015

[8] Leo Moser On the additive completion of sets of integers, Proceedings of Symposia in Pure Mathematics, Volume 8 (1965), pp. 175-180 | Article | MR 175874 | Zbl 0133.29804