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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} -
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C. R. Matthews, L. N. Vaserstein, and B. Weisfeiler, "Congruence properties of Zariski-dense subgroups. I," Proc. London Math. Soc., vol. 48, iss. 3, pp. 514-532, 1984.
@article {MatthewsVasersteinWeisfeiler1984, MRKEY = {0735226},
AUTHOR = {Matthews, C. R. and Vaserstein, L. N. and Weisfeiler, B.},
TITLE = {Congruence properties of {Z}ariski-dense subgroups. {I}},
JOURNAL = {Proc. London Math. Soc.},
FJOURNAL = {Proceedings of the London Mathematical Society. Third Series},
VOLUME = {48},
YEAR = {1984},
NUMBER = {3},
PAGES = {514--532},
ISSN = {0024-6115},
CODEN = {PLMTAL},
MRCLASS = {20G35 (11R37 14G25 14L99)},
MRNUMBER = {0735226},
MRREVIEWER = {Jean-Pierre Wintenberger},
ZBLNUMBER = {0551.20029},
DOI = {10.1112/plms/s3-48.3.514},
} -
[Naud2005]
F. Naud, "Expanding maps on Cantor sets and analytic continuation of zeta functions," Ann. Sci. École Norm. Sup., vol. 38, iss. 1, pp. 116-153, 2005.
@article {Naud2005, MRKEY = {2136484},
AUTHOR = {Naud, Fr{é}d{é}ric},
TITLE = {Expanding maps on {C}antor sets and analytic continuation of zeta functions},
JOURNAL = {Ann. Sci. École Norm. Sup.},
FJOURNAL = {Annales Scientifiques de l'École Normale Supérieure. Quatrième Série},
VOLUME = {38},
YEAR = {2005},
NUMBER = {1},
PAGES = {116--153},
ISSN = {0012-9593},
CODEN = {ASENAH},
MRCLASS = {37C30 (37A45 37B10 37C35 37D35 37D40 37F50)},
MRNUMBER = {2136484},
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[Niederreiter1978]
H. Niederreiter, "Quasi-Monte Carlo methods and pseudo-random numbers," Bull. Amer. Math. Soc., vol. 84, iss. 6, pp. 957-1041, 1978.
@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},
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PAGES = {957--1041},
ISSN = {0002-9904},
MRCLASS = {65C05 (10K15 65C10 68J05)},
MRNUMBER = {0508447},
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}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.
@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},
}