Spiraling spectra of geodesic lines in negatively curved manifolds

TL;DR

This paper introduces a spiraling spectrum for geodesic lines in negatively curved spaces, providing new insights into their asymptotic behavior and applications to Diophantine approximation across various mathematical structures.

Contribution

It defines a new spiraling spectrum concept and proves analogs of classical theorems, extending Diophantine approximation results to complex and Heisenberg group elements.

Findings

Established a precise description of geodesic spiraling behavior.

Proved analogs of Dirichlet, Hall, and Cusick theorems.

Applied results to Diophantine approximation in multiple settings.

Abstract

Given a negatively curved geodesic metric space M, we study the asymptotic penetration behaviour of geodesic lines of M in small neighbourhoods of closed geodesics and of other compact convex subsets of M. We define a spiraling spectrum which gives precise information on the asymptotic spiraling lengths of geodesic lines around these objects. We prove analogs of the theorems of Dirichlet, Hall and Cusick in this context. As a consequence, we obtain Diophantine approximation results of real numbers, complex numbers, or elements of the Heisenberg group by irrational quadratic ones.