
Moscow J. Combin. Number Theory, 2014, Volume 4, Issue 1, Pages 78–117
(Mi mjcnt2)




A strengthening of a theorem of Bourgain–Kontorovich II
D. A. Frolenkov^{a}, I. D. Kan^{b} ^{a} Division of Algebra and Number Theory,
Steklov Mathematical Institute,
Gubkina str., 8, Moscow, Russia 119991
^{b} Department of Number Theory,
Moscow State University,
Moscow, Russia
Abstract:
Zaremba’s conjecture (1971) states that every positive integer number $d$ can be represented as a denominator (continuant) of a finite continued fraction
$\frac bd= [d_1, d_2 ,…, d_k]$, with
all partial quotients $d_1, d_2 ,…, d_k$ being bounded by an absolute constant $A$. Recently (in 2011)
several new theorems concerning this conjecture were proved by Bourgain and Kontorovich. The
easiest of them states that the set of numbers satisfying Zaremba’s conjecture with $A = 50$ has
positive proportion in $\mathbb{N}$. In 2013 we proved this result with $A = 7$. In this paper the same theorem
is proved with $A = 5$.
Funding Agency 
Grant Number 
Russian Foundation for Basic Research 
120131165 140100203 140190002 
Dynasty Foundation 

D.A. Frolenkov was supported by the Dynasty Foundation and by the Russian Foundation for Basic
Research (grants no. 12–01–31165mol_a and no. 14–01–00203A and no. 14–01–90002Bel_a). I.D.Kan was supported by RFFI (grant no. 12–01–00681a). 
Bibliographic databases:
MSC: 11A55, 11L07, 11P55, 11D79 Received: 03.06.2013 Revised: 28.12.2013
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