On the closedness of approximation spectra

TL;DR

This paper proves that various approximation spectra are closed by generalizing Cusick's theorem, utilizing geodesic flow properties in negatively curved manifolds and a key result by Maucourant.

Contribution

It extends the classical Lagrange spectrum's closedness to broader approximation spectra using geometric and dynamical methods.

Findings

Various approximation spectra are closed.

The proof uses geodesic flow penetration properties.

A result of Maucourant is instrumental in the proof.

Abstract

Generalizing Cusick's theorem on the closedness of the classical Lagrange spectrum for the approximation of real numbers by rational ones, we prove that various approximation spectra are closed, using penetration properties of the geodesic flow in cusp neighbourhoods in negatively curved manifolds and a result of Maucourant.