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On the Exact Fourier Dimension of Sets of Well-Approximable Matrices

Thomas Cai, Kyle Hambrook

Abstract

We compute the exact Fourier dimension of the set of $Ψ$-well-approximable $m \times n$ matrices (and the set of $Ψ$-well-approximable numbers) in the homogeneous and inhomogeneous cases for any approximation function $Ψ$ satisfying $\sum_{q \in \mathbb{Z}^n} Ψ(q)^m < \infty$.

On the Exact Fourier Dimension of Sets of Well-Approximable Matrices

Abstract

We compute the exact Fourier dimension of the set of -well-approximable matrices (and the set of -well-approximable numbers) in the homogeneous and inhomogeneous cases for any approximation function satisfying .
Paper Structure (16 sections, 19 theorems, 102 equations)

This paper contains 16 sections, 19 theorems, 102 equations.

Table of Contents

  1. Introduction
  2. Background
  3. Definition of Well-Approximable Matrices
  4. Previous Results
  5. Main Theorem
  6. The Divergence Case
  7. Proof of Proposition \ref{['main-upper-bound']}
  8. Inhomogeneous Lattice Lemma
  9. Countable Stability Lemma
  10. Borel-Cantelli Argument
  11. Proof of Proposition \ref{['main-lower-bound']}
  12. Preliminaries
  13. Periodic Bump Function
  14. Single-Scale Function
  15. Stability Lemma
  16. ...and 1 more sections

Key Result

Theorem 1.4.1

If $\sum_{q \in Q} \Psi(q)^m < \infty$, then $\dim_F E(m,n,Q,\Psi, \theta) = \min\{2s(Q,\Psi), mn\}$.

Theorems & Definitions (32)

  • Theorem 1.4.1
  • Remark 1.4.2
  • Proposition 1.4.3
  • Proposition 1.4.4
  • Example 2.0.1
  • Lemma 2.0.2
  • proof
  • Corollary 2.0.3
  • Lemma 3.1.1
  • Lemma 3.1.2
  • ...and 22 more