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Bull. London Math. Soc. Page 1 of 10 C2010 London Mathematical Society doi:10.1112/blms/bdq021 The number of rational numbers determined by large sets of integers J. Cilleruelo, D. S. Ramana and O. Ramare Abstract When A and B are subsets of the integers in [1, X] and [1, Y ], respectively, with |A| ?X and |B| ?Y , we show that the number of rational numbers expressible as a/b with (a, b) in A ? B is (??)1+XY for any > 0, where the implied constant depends on alone. We then construct examples that show that this bound cannot, in general, be improved to ??XY . We also resolve the natural generalization of our problem to arbitrary subsets C of the integer points in [1, X] ? [1, Y ]. Finally, we apply our results to answer a question of Sarkozy concerning the di?erences of consecutive terms of the product sequence of a given integer sequence. 1. Introduction When A and B are subsets of the positive integers let A/B be the set of all rational numbers expressible as a/b with (a, b) in A?B. Suppose now that A and B are intervals in the integers in [1,X] and [1, Y ] respectively, satisfying |A| ?X and |B| ?Y , where X, Y real numbers at least 1, ?, ?

- rational numbers expressible
- rational numbers
- integers
- c2010 london
- london math
- xy d2
- constant associated
- mathematical society doi

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Published by | profil-urra-2012 |

Reads | 16 |

Language | English |

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