# The hyperbolic geometry of Markov's theorem on Diophantine approximation and quadratic forms

## TL;DR

This paper offers a new proof of Markov's theorem on Diophantine approximation using hyperbolic geometry, connecting geometric concepts with algebraic number theory and quadratic forms.

## Contribution

It introduces a novel geometric proof of Markov's theorem by translating between hyperbolic geometry and number theory, utilizing tools from Teichmüller theory.

## Key findings

- New geometric proof of Markov's theorem
- Connection between hyperbolic geometry and quadratic forms
- Insights into geodesics and tessellations related to Diophantine approximation

## Abstract

Markov's theorem classifies the worst irrational numbers with respect to rational approximation and the indefinite binary quadratic forms whose values for integer arguments stay farthest away from zero. The main purpose of this paper is to present a new proof of Markov's theorem using hyperbolic geometry. The main ingredients are a dictionary to translate between hyperbolic geometry and algebra/number theory, and some very basic tools borrowed from modern geometric Teichm\"uller theory. Simple closed geodesics and ideal triangulations of the modular torus play an important role, and so do the problems: How far can a straight line crossing a triangle stay away from the vertices? How far can it stay away from the vertices of the tessellation generated by the triangle? Definite binary quadratic forms are briefly discussed in the last section.