Partial quotients and representation of rational numbers

Jean Bourgain

doi:10.1016/j.crma.2012.09.002

Logic

Partial quotients and representation of rational numbers

Jean Bourgain

Jean Bourgain ¹

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

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

Abstract (Original language)
Résumé

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.

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.

Received: 2012-07-23
Accepted: 2012-09-04
Published online: 2012-08-01

DOI: 10.1016/j.crma.2012.09.002

Author's affiliations:

Jean Bourgain ¹

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

@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},
     year = {2012},
     publisher = {Elsevier},
     volume = {350},
     number = {15-16},
     doi = {10.1016/j.crma.2012.09.002},
     language = {en},
}

TY  - JOUR
AU  - Jean Bourgain
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
DO  - 10.1016/j.crma.2012.09.002
LA  - en
ID  - CRMATH_2012__350_15-16_727_0
ER  -

%0 Journal Article
%A Jean Bourgain
%T Partial quotients and representation of rational numbers
%J Comptes Rendus. Mathématique
%D 2012
%P 727-730
%V 350
%N 15-16
%I Elsevier
%R 10.1016/j.crma.2012.09.002
%G en
%F CRMATH_2012__350_15-16_727_0

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

References
Cited by

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

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

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

[4] R. Kenyon, private communication.

Cited by Sources:

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

Comments - Policy