A strengthening of a theorem of Bourgain-Kontorovich-III

TL;DR

This paper advances Zaremba's conjecture by proving that a positive proportion of integers can be represented with continued fractions having partial quotients from the set {1, 2, 3, 4, 10}, improving previous bounds.

Contribution

The paper proves that a positive proportion of integers satisfy Zaremba's conjecture using a specific alphabet, extending prior results with A=5 to a new set of partial quotients.

Findings

Positive proportion of integers with partial quotients in {1,2,3,4,10} satisfy Zaremba's conjecture.

Extends previous results from A=5 to a new alphabet set.

Provides new techniques for analyzing continued fraction representations.

Abstract

Zaremba's conjecture (1971) states that every positive integer number can be represented as a denominator (continuant) of a finit continued fraction with all partial quotients 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 2014 the author with D. A. Frolenkov proved this result with A =5. In this paper the same theorem is proved with alphabet {1, 2, 3, 4, 10}