Moscow Journal of Combinatorics and Number Theory
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 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. Frolenkova, I. D. Kanb

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 12-01-3116514-01-0020314-01-90002 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
Revised: 28.12.2013
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