Number Theory
On the Erdős–Turán conjecture
Comptes Rendus. Mathématique, Volume 350 (2012) no. 21-22, pp. 933-935.

Let $N$ be the set of all nonnegative integers. For a set $A⊆N$, let $R(A,n)$ denote the number of solutions $(a,a′)$ of $a+a′=n$ with $a,a′∈A$. 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 $A⊆N$ such that $R(A,n)⩾1$ for all integers $n⩾0$ and the set of n with $R(A,n)=2$ has density one.

Soit $N$ lʼensemble des entiers positifs ou nul. Pour un sous-ensemble $A⊂N$ nous notons $R(A,n)$ le nombre de solutions $(a,a′)∈A2$ 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 $A⊂N$ 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.

Accepted:
Published online:
DOI: 10.1016/j.crma.2012.10.022

Yong-Gao Chen 1

1 School of Mathematical Sciences and Institute of Mathematics, Nanjing Normal University, Nanjing 210023, PR China
@article{CRMATH_2012__350_21-22_933_0,
author = {Yong-Gao Chen},
title = {On the {Erd\H{o}s{\textendash}Tur\'an} conjecture},
journal = {Comptes Rendus. Math\'ematique},
pages = {933--935},
publisher = {Elsevier},
volume = {350},
number = {21-22},
year = {2012},
doi = {10.1016/j.crma.2012.10.022},
language = {en},
}
TY  - JOUR
AU  - Yong-Gao Chen
TI  - On the Erdős–Turán conjecture
JO  - Comptes Rendus. Mathématique
PY  - 2012
SP  - 933
EP  - 935
VL  - 350
IS  - 21-22
PB  - Elsevier
DO  - 10.1016/j.crma.2012.10.022
LA  - en
ID  - CRMATH_2012__350_21-22_933_0
ER  - 
%0 Journal Article
%A Yong-Gao Chen
%T On the Erdős–Turán conjecture
%J Comptes Rendus. Mathématique
%D 2012
%P 933-935
%V 350
%N 21-22
%I Elsevier
%R 10.1016/j.crma.2012.10.022
%G en
%F CRMATH_2012__350_21-22_933_0
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/

[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 $Zm$, 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 $Zm$, Colloq. Math., Volume 104 (2006), pp. 99-103

Cited by Sources:

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