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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, ?, ?

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Rational number Integer

Bulletin of the London Mathematical Society Advance Access published April 18, 2010

Bull. London Math. Soc.Page 1 of 10

C❡ 2010 London Mathematical Society doi:10.1112/blms/bdq021

The number of rational numbers determined by large sets of integers

J.Cilleruelo,D.S.RamanaandO.Ramare´

Abstract WhenAandBare subsets of the integers in [1, X] and [1, Y], respectively, with|A| αXand|B|βY, we show that the number of rational numbers expressible asa/bwith 1+ (a, b) inA×Bis(αβ)XYfor any >0, where the implied constant depends on alone. We then construct examples that show that this bound cannot, in general, be improved toαβXY. We also resolve the natural generalization of our problem to arbitrary subsets Cof the integer points in [1, X]×[1, Y]. Finally, we apply our results to answer a question ofS´ark¨ozyconcerningthediﬀerencesofconsecutivetermsoftheproductsequenceofagiven integer sequence.

1.Introduction WhenAandBare subsets of the positive integers letA/Bbe the set of all rational numbers expressible asa/bwith (a, b) inA×B. Suppose now thatAandBare intervals in the integers in [1, X] and [1, Y] respectively, satisfying|A|αXand|B|βY, whereX,Yreal numbers at least 1,α,βare real numbers in (0,sioniusinvertfeh¨Mbocitaoionrddaplap.A1]anst formula then shows that|A/B| αβXY. Our purpose is to investigate what might be deduced when in place ofintervalswe consider arbitrarysubsetsAandBof the integers in [1, X] and [1, Y] respectively with|A|αX and|B|βY. In this general situation, the main diﬃculty arises from the fact that certain integersdmay have an abnormally large number of multiples in the setsAandB. Further, these integersdare not determined by the conditions onAandB, which are only in terms of their cardinalities. Nevertheless, since the sets under consideration are large, popular heuristics suggest that a non-trivial conclusion should still be accessible. What is pleasing is that we in fact have the following theorem, which is our principal conclusion.

Theorem 1.1.Letαandβbe real numbers in(0,1]and letXandYbe real numbers at least1. WhenAandBare subsets of the integers in[1, X]and[1, Y], respectively, with|A| 1+ αXand|B|βY, we have|A/B| (αβ)XYfor any >0, where the implied constant depends onalone.

Deferring the detailed proof of Theorem 1.1 to Section 2, let us summarize our argument for it with the aid of the following notation. For any integerd1,AandBsubsets of the integers, we writeM(A, B, d) to denote the subset ofA×Bconsisting of all (a, b) in A×Bwith gcd(a, b) =d. We show in Proposition 2.1 that forAandBas in Theorem 2 1.1, we have sup|M(A, B, d)|(1/8)(αβ)XY. Starting from this initial bound, we then d1

Received 30 July 2009; revised 27 November 2009. 2000Mathematics Subject Classiﬁcation11B05 (primary), 11B30 (secondary). The ﬁrst author was supported by Grant CCG08-UAM/ESP-3906 and MTM2008-03880 of the MYCIT, Spain and the third author was supported by the CNRS, France.

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