Logic
Partial quotients and representation of rational numbers
Comptes Rendus. Mathématique, Volume 350 (2012) no. 15-16, pp. 727-730.

It is shown that there is an absolute constant C such that any rational bq]0,1[, (b,q)=1, admits a representation as a finite sum bq=αbαqα where αiai(bαqα)<Clogq and {ai(x)} denotes the sequence of partial quotients of x.

On démontre lʼexistence dʼune constante C telle que tout rationnel bq]0,1[, (b,q)=1, a une représentation comme somme finie bq=αbαqααiai(bαqα)<Clogq et {ai(x)} est la suite des quotients partiels de x.

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DOI: 10.1016/j.crma.2012.09.002
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, Volume 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/

[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.

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