Dimension Functions on the Spectrum over Bounded Geodesics and Applications to Diophantine Approximation

2014/03/27 by Steffen Weil, Weil, Steffen
Mathematics · #Dynamical Systems (math.DS) #FOS: Mathematics #Metric Geometry (math.MG) #Number Theory (math.NT) #math.DS #math.MG #math.NT

paper · pdf · doi:10.48550/arxiv.1403.7015

Comments are welcome! Further improvements and results in new version

arxiv created 2014/09/05 · arxiv updated 2014/09/08

Abstract

The set B of geodesic rays avoiding a suitable obstacle in a complete negatively curved Riemannian manifold determines a spectrum S. While various properties of this spectrum are known, we define and study dimension functions on S in terms of the Hausdorff-dimension of suitable subsets of the set of bounded geodesic rays. We establish estimates on the Hausdorff-dimension of these subsets and thereby obtain non-trivial bounds for the dimension functions. Moreover we discuss the property of B being an absolute winning set, therefore satisfying a remarkable rigidity. Finally, we apply the obtained results to the dimension functions on the spectrum of complex numbers badly approximable by either an imaginary quadratic number field Q(i √(d)) or by quadratic irrational numbers over Q(i √(d)).

Related

• Bounded geodesics in moduli space
• Dimension of Diophantine approximation and some applications in harmonic analysis
• Concentration of dimension in extremal points of left-half lines in the Lagrange spectrum
• Dimension Spectra of Lines
• Concentration of dimension in the Lagrange spectrum