The number of rational numbers determined by large sets of integers

Ramaré, Olivier; Cilleruelo, J.; Ramana, D.

Document type :

Compte-rendu et recension critique d'ouvrage

DOI :

10.1112/blms/bdq021

Title :

The number of rational numbers determined by large sets of integers

Author(s) :

Ramaré, Olivier [Auteur]
Centre National de la Recherche Scientifique [CNRS]
Laboratoire Paul Painlevé - UMR 8524 [LPP]
Cilleruelo, J. [Auteur]
Ramana, D. [Auteur]

Journal title :

Bulletin of the London Mathematical Society

Pages :

517-526

Publisher :

London Mathematical Society

Publication date :

2010-06

ISSN :

0024-6093

HAL domain(s) :

Mathématiques [math]/Théorie des nombres [math.NT]

English abstract : [en]

When $A$ and $B$ are subsets of the integers in $[1, X]$ and $[1, Y ]$, respectively, with $|A| \ge\alpha X$ and $|B|\ge \beta Y$ , we show that the number of rational numbers expressible as $a/b$ with $(a, b)$ in $A \times$ ...
Show more >When $A$ and $B$ are subsets of the integers in $[1, X]$ and $[1, Y ]$, respectively, with $|A| \ge\alpha X$ and $|B|\ge \beta Y$ , we show that the number of rational numbers expressible as $a/b$ with $(a, b)$ in $A \times$ B is $(\alpha\beta)^{1+\epsilon} XY$ for any $\epsilon> 0$, where the implied constant depends on $\epsilon$ alone. We then construct examples that show that this bound cannot, in general, be improved to $\alpha\beta XY$. We also resolve the natural generalization of our problem to arbitrary subsets $C$ of the integer points in $[1, X] \times [1, Y ]$. Finally, we apply our results to answer a question of Sárközy concerning the differences of consecutive terms of the product sequence of a given integer sequence.Show less >

Language :

Anglais

Popular science :

Non

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