On Zaremba's Conjecture

TL;DR

This paper confirms Zaremba's conjecture for a set of integers with density one, showing that most integers can be represented as denominators of finite continued fractions with bounded partial quotients.

Contribution

The paper proves Zaremba's conjecture for a set of density one, advancing understanding of continued fractions and number theory.

Findings

Confirmed Zaremba's conjecture for a density one set of integers

Most integers can be expressed as denominators of bounded partial quotient continued fractions

Provides new insights into the distribution of such fractions

Abstract

Zaremba's 1971 conjecture predicts that every integer appears as the denominator of a finite continued fraction whose partial quotients are bounded by an absolute constant. We confirm this conjecture for a set of density one.