# Approximation of the Lagrange and Markov spectra

## TL;DR

This paper presents a polynomial time algorithm for approximating the complex Lagrange spectrum and extends the method to the Markov spectrum, both related to diophantine approximation and quadratic forms.

## Contribution

It introduces a novel polynomial time algorithm for approximating the Lagrange spectrum and extends this approach to the Markov spectrum.

## Key findings

- Algorithm approximates the Lagrange spectrum in Hausdorff distance
- Extension of the algorithm to the Markov spectrum
- Efficient approximation method for spectra related to quadratic forms

## Abstract

The (classical) Lagrange spectrum is a closed subset of the positive real numbers defined in terms of diophantine approximation. Its structure is quite involved. This article describes a polynomial time algorithm to approximate it in Hausdorff distance. It also extends to approximate the Markov spectrum related to infimum of binary quadratic forms.