On the Erdős–Turán conjecture

Yong-Gao Chen

doi:10.1016/j.crma.2012.10.022

Number Theory

On the Erdős–Turán conjecture
[Sur la conjecture dʼErdös–Turán]

Présenté par : the Editorial Board

Yong-Gao Chen ¹

¹ School of Mathematical Sciences and Institute of Mathematics, Nanjing Normal University, Nanjing 210023, PR China

Comptes Rendus. Mathématique, Volume 350 (2012) no. 21-22, pp. 933-935.

Résumé
Abstract

Soit $N$ lʼensemble des entiers positifs ou nul. Pour un sous-ensemble $A \subset N$ nous notons $R (A, n)$ le nombre de solutions $(a, a^{'}) \in A^{2}$ de $a + a^{'} = n$ . La célèbre conjecture dʼErdös–Turán affirme que si $R (A, n) ⩾ 1$ pour tout entier $n ⩾ 0$ , alors $R (A, n)$ nʼest pas borné. Nous montrons dans cette Note quʼil existe un sous-ensemble $A \subset N$ tel que $R (A, n) ⩾ 1$ pour tout entier $n ⩾ 0$ et tel que lʼensemble des n satisfaisant $R (A, n) = 2$ soit de densité un.

Let $N$ be the set of all nonnegative integers. For a set $A \subseteq N$ , let $R (A, n)$ denote the number of solutions $(a, a^{'})$ of $a + a^{'} = n$ with $a, a^{'} \in A$ . The well known Erdős–Turán conjecture says that if $R (A, n) ⩾ 1$ for all integers $n ⩾ 0$ , then $R (A, n)$ is unbounded. In this Note, the following result is proved: There is a set $A \subseteq N$ such that $R (A, n) ⩾ 1$ for all integers $n ⩾ 0$ and the set of n with $R (A, n) = 2$ has density one.

Reçu le : 2012-06-11
Accepté le : 2012-10-25
Publié le : 2012-11-01

DOI : 10.1016/j.crma.2012.10.022

Affiliations des auteurs :

Yong-Gao Chen ¹

¹ School of Mathematical Sciences and Institute of Mathematics, Nanjing Normal University, Nanjing 210023, PR China

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     title = {On the {Erd\H{o}s{\textendash}Tur\'an} conjecture},
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Yong-Gao Chen. On the Erdős–Turán conjecture. Comptes Rendus. Mathématique, Volume 350 (2012) no. 21-22, pp. 933-935. doi : 10.1016/j.crma.2012.10.022. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2012.10.022/

Bibliographie
Cité par

[1] P. Borwein; S. Choi; F. Chu An old conjecture of Erdős–Turán on additive bases, Math. Comp., Volume 75 (2006), pp. 475-484

[2] Y.-G. Chen The analogue of Erdős–Turán conjecture in $Z_{m}$ , J. Number Theory, Volume 128 (2008), pp. 2573-2581

[3] Y.-G. Chen; Q.-H. Yang Ruzsaʼs theorem on Erdős and Turán conjecture, European J. Combin., Volume 34 (2013), pp. 410-413 | DOI

[4] P. Erdős; P. Turán On a problem of Sidon in additive number theory, and on some related problems, J. London Math. Soc., Volume 16 (1941), pp. 212-215

[5] G. Grekos; L. Haddad; C. Helou; J. Pihko On the Erdős–Turán conjecture, J. Number Theory, Volume 102 (2003), pp. 339-352

[6] G.H. Hardy; E.M. Wright An Introduction to the Theory of Numbers, Oxford Univ. Press, 1979

[7] M.B. Nathanson Minimal bases and powers of 2, Acta Arith., Volume 49 (1988), pp. 525-532

[8] M.B. Nathanson Unique representation bases for integers, Acta Arith., Volume 108 (2003), pp. 1-8

[9] I.Z. Ruzsa A just basis, Monatsh. Math., Volume 109 (1990), pp. 145-151

[10] M. Tang A note on a result of Ruzsa, II, Bull. Aust. Math. Soc., Volume 82 (2010), pp. 340-347

[11] M. Tang; Y.-G. Chen A basis of $Z_{m}$ , Colloq. Math., Volume 104 (2006), pp. 99-103

Cité par Sources :

^☆ This work was supported by the National Natural Science Foundation of China, Grant No. 11071121.

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