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 AN, let R(A,n) denote the number of solutions (a,a) of a+a=n with a,aA. The well known Erdős–Turán conjecture says that if R(A,n)1 for all integers n0, then R(A,n) is unbounded. In this Note, the following result is proved: There is a set AN such that R(A,n)1 for all integers n0 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 AN 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 n0, alors R(A,n) nʼest pas borné. Nous montrons dans cette Note quʼil existe un sous-ensemble AN tel que R(A,n)1 pour tout entier n0 et tel que lʼensemble des n satisfaisant R(A,n)=2 soit de densité un.

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DOI: 10.1016/j.crma.2012.10.022
Chen, Yong-Gao 1

1 School of Mathematical Sciences and Institute of Mathematics, Nanjing Normal University, Nanjing 210023, PR China
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Chen, Yong-Gao. 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. http://www.numdam.org/articles/10.1016/j.crma.2012.10.022/

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