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Diophantine approximation in metric space

Jonathan M. Fraser, Henna Koivusalo, Felipe A. Ramirez

Abstract

Diophantine approximation is traditionally the study of how well real numbers are approximated by rationals. We propose a model for studying Diophantine approximation in an arbitrary totally bounded metric space where the rationals are replaced with a countable hierarchy of `well-spread' points, which we refer to as abstract rationals. We prove various Jarnik-Besicovitch type dimension bounds and investigate their sharpness.

Diophantine approximation in metric space

Abstract

Diophantine approximation is traditionally the study of how well real numbers are approximated by rationals. We propose a model for studying Diophantine approximation in an arbitrary totally bounded metric space where the rationals are replaced with a countable hierarchy of `well-spread' points, which we refer to as abstract rationals. We prove various Jarnik-Besicovitch type dimension bounds and investigate their sharpness.

Paper Structure

This paper contains 10 sections, 10 theorems, 32 equations.

Table of Contents

  1. Introduction
  2. Diophantine approximation in a metric space
  3. Dimension theory
  4. Main results
  5. General bounds
  6. Sharpness of general bounds
  7. Sharpness of upper bound
  8. Sharpness of lower bound
  9. The Ahlfors--David regular case
  10. Dirichlet spectrum of Ahlfors--David regular spaces

Key Result

Theorem 2.1

For all $t \geqslant 1$ and, if $F$ is compact,

Theorems & Definitions (11)

  • Theorem 2.1
  • Proposition 2.2
  • Proposition 2.3
  • Proposition 2.4
  • Proposition 2.5
  • Theorem 2.7
  • Proposition 3.1
  • Remark
  • Proposition 3.2
  • Proposition 3.3
  • ...and 1 more