Metric Diophantine Approximation: Aspects of Recent Work

2016/09/30 by Victor Beresnevich, Felipe A. Ramírez, Sanju Velani · 2 citations
Engineering · Mathematics · Physics and Astronomy · #Advanced Mathematical Theories and Applications #Algebra over a field #Analytic Number Theory Research #Diophantine approximation #Diophantine equation #Engineering #Mathematical Dynamics and Fractals #Mathematics #Metric (unit) #Operations management #Physics #Pure mathematics #Thermodynamics #Work (physics)

paper · doi:10.1017/9781316402696.002

openalex publication_date 2016/09/30 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/03

Abstract

In these notes, we begin by recalling aspects of the classical theory of metric Diophantine approximation; such as theorems of Khintchine, Jarnik, Duffin-Schaeffer and Gallagher. We then describe recent strengthening of various classical statements as well as recent developments in the area of Diophantine approximation on manifolds. The latter includes the well approximable, the badly approximable and the inhomogeneous aspects.

Citations

• Open Diophantine Problems
• Badly approximable points on planar curves and winning
• Ubiquity and a general logarithm law for geodesics
• Classical metric Diophantine approximation revisited
• Khintchine’s problem in metric Diophantine approximation
• Flows on Homogeneous Spaces and Diophantine Approximation on Manifolds
• On irregularities of distribution
• A Mass Transference Principle and the Duffin–Schaeffer conjecture for Hausdorff measures
• Weighted badly approximable vectors and games
• Diophantine approximation and the geometry of limit sets in Gromov hyperbolic metric spaces

Cited by

• Winning and nullity of inhomogeneous bad
• Vojtěch Jarník [wikipedia]

Related

• Metric Diophantine Approximation: aspects of recent work
• Classical metric Diophantine approximation revisited
• A century of metric Diophantine approximation and half a decade since Koukoulopoulos-Maynard
• Metric Diophantine approximation: The Khintchine--Groshev theorem for non-degenerate manifolds
• A note on well distributed sequences