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.
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.
Accepté le :
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@article{CRMATH_2012__350_15-16_727_0, author = {Bourgain, Jean}, 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}, url = {http://www.numdam.org/articles/10.1016/j.crma.2012.09.002/} }
TY - JOUR AU - Bourgain, Jean TI - Partial quotients and representation of rational numbers JO - Comptes Rendus. Mathématique PY - 2012 SP - 727 EP - 730 VL - 350 IS - 15-16 PB - Elsevier UR - http://www.numdam.org/articles/10.1016/j.crma.2012.09.002/ DO - 10.1016/j.crma.2012.09.002 LA - en ID - CRMATH_2012__350_15-16_727_0 ER -
%0 Journal Article %A Bourgain, Jean %T Partial quotients and representation of rational numbers %J Comptes Rendus. Mathématique %D 2012 %P 727-730 %V 350 %N 15-16 %I Elsevier %U http://www.numdam.org/articles/10.1016/j.crma.2012.09.002/ %R 10.1016/j.crma.2012.09.002 %G en %F CRMATH_2012__350_15-16_727_0
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/
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[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.