Comptes Rendus
Number Theory
On Zarembaʼs conjecture
Comptes Rendus. Mathématique, Volume 349 (2011) no. 9-10, pp. 493-495.

It is shown that there is a constant A and a density one subset S of the positive integers such that, for each qS, there is some 1p<q, (p,q)=1, so that pq has all its partial quotients bounded by A.

On montre quʼil existe une constante A et un sous-ensemble S des entiers positifs de densité un, tel que pour tout qS il y a un entier 1p<q,(p,q)=1 pour lequel les quotients partiels de pq sont bornés par A.

Received:
Accepted:
Published online:
DOI: 10.1016/j.crma.2011.03.023
Jean Bourgain 1; Alex Kontorovich 2

1 IAS, Princeton, NJ 08540, USA
2 Stony Brook University, Stony Brook, NY 11794, USA
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Jean Bourgain; Alex Kontorovich. On Zarembaʼs conjecture. Comptes Rendus. Mathématique, Volume 349 (2011) no. 9-10, pp. 493-495. doi : 10.1016/j.crma.2011.03.023. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2011.03.023/

[1] Jean Bourgain, Alex Gamburd, Peter Sarnak, Generalization of Selbergʼs theorem and Selbergʼs sieve, 2009, preprint.

[2] J. Bourgain; A. Kontorovich On representations of integers in thin subgroups of SL(2,Z), GAFA, Volume 20 (2010) no. 5, pp. 1144-1174

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[7] Harald Niederreiter Quasi-Monte Carlo methods and pseudo-random numbers, Bull. Amer. Math. Soc., Volume 84 (1978) no. 6, pp. 957-1041

[8] Harald Niederreiter Dyadic fractions with small partial quotients, Monatsh. Math., Volume 101 (1986) no. 4, pp. 309-315

[9] S.K. Zaremba La méthode des “bons treillis” pour le calcul des intégrales multiples, Univ. Montreal, Montreal, Que., 1971, Academic Press, New York (1972), pp. 39-119

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