On Zaremba's Conjecture

TL;DR

This paper proves that a density one subset of positive integers can be represented with fractions having bounded partial quotients, advancing understanding of Zaremba's conjecture.

Contribution

It establishes the existence of a large subset of integers for which Zaremba's conjecture holds with bounded partial quotients.

Findings

Existence of a density one subset S of positive integers.

For all q in S, there exists p coprime to q with p/q having bounded partial quotients.

Progress towards Zaremba's conjecture with a large subset of integers.

Abstract

It is shown that there is a constant A and a density one subset S of the positive integers, such that for all q in S there is some 1<=p<q, (p, q)=1, so that p/q has all its partial quotients bounded by A.