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On a Kurzweil type theorem via ubiquity

Taehyeong Kim

Abstract

Kurzweil's theorem ('55) is concerned with zero-one laws for well approximable targets in inhomogeneous Diophantine approximation under the badly approximable assumption. In this article, we prove the divergent part of a Kurzweil type theorem via a suitable construction of ubiquitous systems when the badly approximable assumption is relaxed. Moreover, we also discuss some counterparts of Kurzweil's theorem.

On a Kurzweil type theorem via ubiquity

Abstract

Kurzweil's theorem ('55) is concerned with zero-one laws for well approximable targets in inhomogeneous Diophantine approximation under the badly approximable assumption. In this article, we prove the divergent part of a Kurzweil type theorem via a suitable construction of ubiquitous systems when the badly approximable assumption is relaxed. Moreover, we also discuss some counterparts of Kurzweil's theorem.
Paper Structure (8 sections, 12 theorems, 51 equations)

This paper contains 8 sections, 12 theorems, 51 equations.

Table of Contents

  1. Introduction
  2. Convergent part: a warm up
  3. Preliminaries for divergent part
  4. Ubiquity systems
  5. Transference principle
  6. Weyl type uniform distribution
  7. Proof of Theorem \ref{['thm_main']}
  8. Proof of Theorem \ref{['thm_counter']}

Key Result

Theorem 1.1

Kur55 If $A\in M_{m,n}(\mathbb{R})$ is badly approximable, then for any decreasing $\psi:\mathbb{R}^+ \to \mathbb{R}^+$ we have Here and hereafter, $|\cdot|$ stands for Lebesgue measure on $\mathbb{R}^m$.

Theorems & Definitions (24)

  • Theorem 1.1
  • Remark 1.2
  • Theorem 1.3
  • Corollary 1.4
  • Remark 1.5
  • Corollary 1.6
  • proof
  • Theorem 1.7
  • Remark 1.8
  • Lemma 2.1: Hausdorff-Cantelli
  • ...and 14 more