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
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.