no. 9-10
Number Theory
On Zarembaʼs conjecture
[Sur une conjecture de Zaremba]

Présenté par : Jean Bourgain

Bourgain, Jean1 ; Kontorovich, Alex2
1 IAS, Princeton, NJ 08540, USA
2 Stony Brook University, Stony Brook, NY 11794, USA
Comptes Rendus. Mathématique, Tome 349 (2011) no. 9-10, pp. 493-495.

On montre quʼil existe une constante A et un sous-ensemble S des entiers positifs de densité un, tel que pour tout q∈S il y a un entier 1⩽p<q,(p,q)=1 pour lequel les quotients partiels de pq 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∈S, there is some 1⩽p<q, (p,q)=1, so that pq has all its partial quotients bounded by A.

Reçu le :
Accepté le :
Publié le :
DOI : 10.1016/j.crma.2011.03.023
Bourgain, Jean 1 ; Kontorovich, Alex 2

1 IAS, Princeton, NJ 08540, USA
2 Stony Brook University, Stony Brook, NY 11794, USA 
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Bourgain, Jean; Kontorovich, Alex. On Zarembaʼs conjecture. Comptes Rendus. Mathématique, Tome 349 (2011) no. 9-10, pp. 493-495. doi : 10.1016/j.crma.2011.03.023. http://www.numdam.org/articles/10.1016/j.crma.2011.03.023/

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

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

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

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

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

[9] Zaremba, S.K. 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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