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On The Hausdorff Dimension Of Two Dimensional Badly Approximable Vector

Yi Lou

Abstract

Let $Ψ=(ψ_1,\dots,ψ_m)$ be an approximation function and denote by $\mathcal{A}_m(Ψ)$ the set of $Ψ$-approximable vectors in $[0,1]^m$. The associated set of weighted $Ψ$-badly approximable vectors is defined by $$\mathcal{B}_m(Ψ)=\mathcal{A}_m(Ψ)\setminus\bigcap \limits_{0<c<1}\mathcal{A}_m(cΨ).$$ In this paper, we study the Hausdorff dimension of $\mathcal{B}_m(Ψ)$ in the weighted power-law setting $ψ_i(q)=q^{-τ_i}$. Our main result establishes a sharp local Hausdorff dimension formula for $\mathcal{B}_2(Ψ_{\boldsymbolτ})$ when $\boldsymbolτ=(τ_1,τ_2)$ satisfies $τ_1\geqτ_2>0$ and $τ_1+τ_2>1$. We show that for any ball $B\subseteq[0,1]^2$, $$\dim_{\mathcal{H}}(B\cap \mathcal{B}_2(Ψ_{\boldsymbol τ})) = \dim_{\mathcal{H}} \mathcal{A}_2(Ψ_{\boldsymbol τ}) =\min \left\{\frac{3+τ_1-τ_2}{1+τ_1},\frac{3}{1+τ_2}\right\}.$$ The proof extends the Cantor-type construction and mass distribution arguments of Koivusalo, Levesley, Ward, and Zhang from the unweighted to the weighted setting, and is independent of recent results on weighted exact approximation.

On The Hausdorff Dimension Of Two Dimensional Badly Approximable Vector

Abstract

Let be an approximation function and denote by the set of -approximable vectors in . The associated set of weighted -badly approximable vectors is defined by In this paper, we study the Hausdorff dimension of in the weighted power-law setting . Our main result establishes a sharp local Hausdorff dimension formula for when satisfies and . We show that for any ball , The proof extends the Cantor-type construction and mass distribution arguments of Koivusalo, Levesley, Ward, and Zhang from the unweighted to the weighted setting, and is independent of recent results on weighted exact approximation.
Paper Structure (5 sections, 15 theorems, 77 equations)

This paper contains 5 sections, 15 theorems, 77 equations.

Table of Contents

  1. Introduction
  2. Step 1: The Set $\mathcal{C}^{\boldsymbol \tau}(N)$
  3. Step 2: The Set of Leading Rationals
  4. Step 3: Construction of $\mathcal{D}_{\epsilon}(B)$
  5. Step 4: Construction of Mass Distribution on $\mathcal{D}_\epsilon(B)$

Key Result

Theorem 1.1

Let $a_i\geq 1$ for $1\leq i \leq d$ and $a_1+\cdots+a_m=1$. For any $\mathbf{x}=(x_1,\cdots,x_m)\in \mathbb{R}^m$, there exist infinitely many integers $p_1,\cdots,p_m,q$ such that

Theorems & Definitions (27)

  • Theorem 1.1: Dirichlet's Theorem
  • Theorem 1.2
  • Remark 1.3
  • Proposition 1.4: Mass Distribution Principle
  • proof
  • Lemma 2.1: Simplex Lemma
  • Lemma 2.2
  • proof
  • Lemma 2.3
  • proof
  • ...and 17 more