On  Zaremba’s conjecture

Jean Bourgain; Alex Kontorovich

doi:https://doi.org/10.4007/annals.2014.180.1.3

Abstract

Zaremba’s 1971 conjecture predicts that every integer appears as the denominator of a finite continued fraction whose partial quotients are bounded by an absolute constant. We confirm this conjecture for a set of density one.

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@article {BourgainGamburd2008, MRKEY = {2415383}, AUTHOR = {Bourgain, Jean and Gamburd, Alex}, TITLE = {Uniform expansion bounds for {C}ayley graphs of {${\rm SL}\sb 2(\Bbb F\sb p)$}}, JOURNAL = {Ann. of Math.}, FJOURNAL = {Annals of Mathematics. Second Series}, VOLUME = {167}, YEAR = {2008}, NUMBER = {2}, PAGES = {625--642}, ISSN = {0003-486X}, CODEN = {ANMAAH}, MRCLASS = {20F65 (05C25 05E15 11B30 20G40)}, MRNUMBER = {2415383}, MRREVIEWER = {Ben Joseph Green}, ZBLNUMBER = {1216.20042}, DOI = {10.4007/annals.2008.167.625}, }
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[Niederreiter1978] $Go to document$ H. Niederreiter, "Quasi-Monte Carlo methods and pseudo-random numbers," Bull. Amer. Math. Soc., vol. 84, iss. 6, pp. 957-1041, 1978.

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@article {Niederreiter1978, MRKEY = {0508447}, AUTHOR = {Niederreiter, Harald}, TITLE = {Quasi-{M}onte {C}arlo methods and pseudo-random numbers}, JOURNAL = {Bull. Amer. Math. Soc.}, FJOURNAL = {Bulletin of the American Mathematical Society}, VOLUME = {84}, YEAR = {1978}, NUMBER = {6}, PAGES = {957--1041}, ISSN = {0002-9904}, MRCLASS = {65C05 (10K15 65C10 68J05)}, MRNUMBER = {0508447}, MRREVIEWER = {George Marsaglia}, ZBLNUMBER = {0404.65003}, DOI = {10.1090/S0002-9904-1978-14532-7}, }xeucli
[Zaremba1972] S. K. Zaremba, "La méthode des “bons treillis” pour le calcul des intégrales multiples," in Applications of Number Theory to Numerical Analysis, New York: Academic Press, 1972, pp. 39-119.

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@incollection {Zaremba1972, MRKEY = {0343530}, AUTHOR = {Zaremba, S. K.}, TITLE = {La méthode des ``bons treillis'' pour le calcul des intégrales multiples}, BOOKTITLE = {Applications of Number Theory to Numerical Analysis}, VENUE={{P}roc. {S}ympos., {U}niv. {M}ontreal, {M}ontreal, {Q}ue., 1971}, PAGES = {39--119}, PUBLISHER = {Academic Press}, ADDRESS = {New York}, YEAR = {1972}, MRCLASS = {65C05 (10K30)}, MRNUMBER = {0343530}, MRREVIEWER = {Jean-Pierre Duport}, ZBLNUMBER = {0246.65009}, }

On Zaremba’s conjecture

Abstract

Authors