On the classificaction of irrational numbers

TL;DR

This paper explores the relationship between the arithmetic and dynamical classifications of irrational numbers, revealing equivalences and conditions linking Diophantine properties and Lyapunov exponents under the Gauss map.

Contribution

It establishes new connections between Diophantine classifications and dynamical properties of irrational numbers via the Gauss map, including conditions involving Lyapunov exponents.

Findings

Equivalence between Diophantine classes and approximation speeds

Irrational numbers with finite Lyapunov exponent satisfy Diophantine conditions

Dynamical and arithmetic classifications are closely linked

Abstract

In this notes we make a comparison between the arithmetic properties of irrational numbers and their dynamical properties under the Gauss map. We show some equivalences between different classifications of irrational numbers such as the Diophantine classes and numbers admitting approximations by rational numbers at a given 'speed'. We also show that irrational numbers with finite upper Lyapunov exponent for the Gauss map satisfy a Diophantine condition.