The number of rational numbers determined by large sets of integers
Résumé
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.
Domaines
Théorie des nombres [math.NT]Origine | Fichiers produits par l'(les) auteur(s) |
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