Partial quotients and representation of rational numbers

Bourgain, Jean

doi:10.1016/j.crma.2012.09.002

Logic

Partial quotients and representation of rational numbers
[Quotients partiels et représentation des nombres rationnels]

Présenté par : Jean Bourgain

Bourgain, Jean ¹

¹ School of Mathematics, Institute for Advanced Study, 1 Einstein Drive, Princeton, NJ 08540, USA

Comptes Rendus. Mathématique, Tome 350 (2012) no. 15-16, pp. 727-730.

Résumé
Abstract

On démontre lʼexistence dʼune constante C telle que tout rationnel $\frac{b}{q} \in] 0, 1 [$ , $(b, q) = 1$ , a une représentation comme somme finie $\frac{b}{q} = \sum_{α} \frac{b_{α}}{q_{α}}$ où $\sum_{α} \sum_{i} a_{i} (\frac{b_{α}}{q_{α}}) < C \log q$ et ${a_{i} (x)}$ est la suite des quotients partiels de x.

It is shown that there is an absolute constant C such that any rational $\frac{b}{q} \in] 0, 1 [$ , $(b, q) = 1$ , admits a representation as a finite sum $\frac{b}{q} = \sum_{α} \frac{b_{α}}{q_{α}}$ where $\sum_{α} \sum_{i} a_{i} (\frac{b_{α}}{q_{α}}) < C \log q$ and ${a_{i} (x)}$ denotes the sequence of partial quotients of x.

Reçu le : 2012-07-23
Accepté le : 2012-09-04
Publié le : 2012-08-01

DOI : 10.1016/j.crma.2012.09.002

Affiliations des auteurs :

Bourgain, Jean ¹

¹ School of Mathematics, Institute for Advanced Study, 1 Einstein Drive, Princeton, NJ 08540, USA

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Bourgain, Jean. Partial quotients and representation of rational numbers. Comptes Rendus. Mathématique, Tome 350 (2012) no. 15-16, pp. 727-730. doi : 10.1016/j.crma.2012.09.002. http://www.numdam.org/articles/10.1016/j.crma.2012.09.002/

Bibliographie
Cité par

[1] J. Bourgain, K. Kontorovich, On Zarembaʼs conjecture, preprint, 2011, . | arXiv

[2] Bourgain, J.; Gamburd, A.; Sarnak, P. Generalization of Selbergʼs 3/16 theorem and affine sieve, Acta Math., Volume 207 (2011) no. 2, pp. 255-290

[3] Hall, M. On the sum and product of continued fractions, Annals of Math., Volume 48 (1947) no. 4

[4] R. Kenyon, private communication.

Cité par Sources :

^☆ The research was partially supported by NSF grants DMS-0808042 and DMS-0835373.