A reinforcement of the Bourgain-Kontorovich's theorem by elementary methods II

TL;DR

This paper proves that a positive proportion of integers satisfy Zaremba's conjecture with partial quotients bounded by 5, using elementary methods, improving previous bounds established by Bourgain and Kontorovich.

Contribution

It demonstrates that elementary methods can establish Zaremba's conjecture for A=5, improving the bound from A=50.

Findings

Positive proportion of integers satisfy Zaremba's conjecture with A=5

Elementary methods suffice for this proof

Improves previous bounds from A=50 to A=5

Abstract

Zaremba's conjecture (1971) states that every positive integer number $d$ can be represented as a denominator (continuant) of a finite continued fraction $\frac{b}{d}=[d_1,d_2,...,d_{k}],$ 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 $\N.$ In this paper,using only elementary methods, the same theorem is proved with A=5.