It is shown that there is an absolute constant C such that any rational , , admits a representation as a finite sum where and denotes the sequence of partial quotients of x.
On démontre lʼexistence dʼune constante C telle que tout rationnel , , a une représentation comme somme finie où et est la suite des quotients partiels de x.
Accepted:
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Jean Bourgain 1
@article{CRMATH_2012__350_15-16_727_0, author = {Jean Bourgain}, title = {Partial quotients and representation of rational numbers}, journal = {Comptes Rendus. Math\'ematique}, pages = {727--730}, publisher = {Elsevier}, volume = {350}, number = {15-16}, year = {2012}, doi = {10.1016/j.crma.2012.09.002}, language = {en}, }
Jean Bourgain. 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. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2012.09.002/
[1] J. Bourgain, K. Kontorovich, On Zarembaʼs conjecture, preprint, 2011, . | arXiv
[2] Generalization of Selbergʼs 3/16 theorem and affine sieve, Acta Math., Volume 207 (2011) no. 2, pp. 255-290
[3] 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.
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