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Article Dans Une Revue Bulletin of the London Mathematical Society Année : 2010

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.
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Dates et versions

hal-02572776 , version 1 (13-05-2020)

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Citer

Olivier Ramaré, J. Cilleruelo, D. S Ramana. The number of rational numbers determined by large sets of integers. Bulletin of the London Mathematical Society, 2010, 42 (3), pp.517-526. ⟨10.1112/blms/bdq021⟩. ⟨hal-02572776⟩
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