A strengthening of a theorem of Bourgain-Kontorovich-IV

TL;DR

This paper improves a key theorem related to Zaremba's conjecture by reducing the bound on partial quotients from 5 to 4, showing a larger set of numbers satisfy the conjecture.

Contribution

The paper provides a proof that the set of numbers satisfying Zaremba's conjecture with A=4 has positive proportion, strengthening previous results with higher bounds.

Findings

Proves the set with A=4 has positive density among natural numbers.

Reduces the partial quotient bound from 5 to 4 in Zaremba's conjecture.

Extends the applicability of Bourgain-Kontorovich's theorem.

Abstract

Zaremba's conjecture (1971) states that every positive integer number d can be represented as a denominator of a finite continued fraction b/d = [d1,d2,...,dk], with all partial quotients d1,d2,...,dk being bounded by an absolute constant A. Several new theorems concerning this conjecture were proved by Bourgain and Kontorovich in 2011. The easiest of them states that the set of numbers satisfying Zaremba's conjecture with A = 50 has positive proportion in natural numbers. In 2014 I. D. Kan and D. A. Frolenkov proved this result with A = 5. In this paper the same theorem is proved with A = 4.