Geodesic
A strengthening of a~theorem of Bourgain and Kontorovich.~IV
Izvestiya. Mathematics , Tome 80 (2016) no. 6, pp. 1094-1117

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We prove that the denominators of finite continued fractions all of whose partial quotients belong to the alphabet $\{1,2,3,4\}$ form a set of positive density. The analogous theorem was known earlier only for alphabets of larger cardinality. The first result of this kind was obtained in 2011 for the alphabet $\{1,2,\dots,50\}$ by Bourgain and Kontorovich. In 2013, the present author, together with Frolenkov, proved the corresponding theorem for the alphabet $\{1,2,3,4,5\}$. A 2014 result of the present author dealt with the alphabet $\{1,2,3,4,10\}$.
Keywords: continued fraction, trigonometric sum
Mots-clés : continuant, Zaremba's conjecture. 
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     author = {I. D. Kan},
     title = {A strengthening of a~theorem of {Bourgain} and {Kontorovich.~IV}},
     journal = {Izvestiya. Mathematics },
     pages = {1094--1117},
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     year = {2016},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/IM2_2016_80_6_a5/}
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I. D. Kan. A strengthening of a~theorem of Bourgain and Kontorovich.~IV. Izvestiya. Mathematics , Tome 80 (2016) no. 6, pp. 1094-1117. http://geodesic.mathdoc.fr/item/IM2_2016_80_6_a5/