On Zarembaʼs conjecture

Jean Bourgain; Alex Kontorovich

doi:10.1016/j.crma.2011.03.023

Number Theory

On Zarembaʼs conjecture
[Sur une conjecture de Zaremba]

Présenté par : Jean Bourgain

Jean Bourgain ¹ ; Alex Kontorovich ²

¹ IAS, Princeton, NJ 08540, USA
² Stony Brook University, Stony Brook, NY 11794, USA

Comptes Rendus. Mathématique, Volume 349 (2011) no. 9-10, pp. 493-495.

Résumé
Abstract

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

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

Reçu le : 2011-03-10
Accepté le : 2011-03-28
Publié le : 2011-04-14

DOI : 10.1016/j.crma.2011.03.023

Affiliations des auteurs :

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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     author = {Jean Bourgain and Alex Kontorovich},
     title = {On {Zaremba's} conjecture},
     journal = {Comptes Rendus. Math\'ematique},
     pages = {493--495},
     publisher = {Elsevier},
     volume = {349},
     number = {9-10},
     year = {2011},
     doi = {10.1016/j.crma.2011.03.023},
     language = {en},
}

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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/

Bibliographie
Cité par

[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

[3] J. Bourgain; A. Kontorovich; P. Sarnak Sector estimates for hyperbolic isometries, GAFA, Volume 20 (2010) no. 5, pp. 1175-1200

[4] Doug Hensley The distribution of badly approximable numbers and continuants with bounded digits, Quebec, PQ, 1987, de Gruyter, Berlin (1989), pp. 371-385

[5] Doug Hensley Continued fraction Cantor sets, Hausdorff dimension, and functional analysis, J. Number Theory, Volume 40 (1992) no. 3, pp. 336-358

[6] Curtis T. McMullen Uniformly Diophantine numbers in a fixed real quadratic field, Compos. Math., Volume 145 (2009) no. 4, pp. 827-844

[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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