On the closedness of approximation spectra

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 o w in cusp neighbourhoods in negatively curved manifolds and a result of Maucourant [Mau]. 1 The approximation constant of an irrational real number x by rational numbers is c(x) = lim inf