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Diophantine approximation and the Mass Transference Principle: incorporating the unbounded setup

Bing Li, Lingmin Liao, Baowei Wnag, Sanju Velani, Evgeniy Zorin

Abstract

We develop the Mass Transference Principle for rectangles of Wang \& Wu (Math. Ann. 2021) to incorporate the `unbounded' setup; that is, when along some direction the lower order (at infinity) of the side lengths of the rectangles under consideration is infinity. As applications, we obtain the Hausdorff dimension of naturally occurring $\limsup$ sets within the classical framework of simultaneous Diophantine approximation and the dynamical framework of shrinking target problems. For instance, concerning the former, for $τ>0$, let $S(τ)$ denote the set of $(x_1,x_2)\in \mathbb{R}^2$ simultaneously satisfying the inequalities $\|q x_1 \| \, < \, q^{-τ} $ and $ \|q x_2 \| \, < \, e^{-q}$ for infinitely many $q \in \mathbb{N}$. Then, the `unbounded' Mass Transference Principle enables us to show that $\dim_{\rm H} S(τ) \, = \, \min \big\{ 1, 3/(1+τ) \big\} \, $.

Diophantine approximation and the Mass Transference Principle: incorporating the unbounded setup

Abstract

We develop the Mass Transference Principle for rectangles of Wang \& Wu (Math. Ann. 2021) to incorporate the `unbounded' setup; that is, when along some direction the lower order (at infinity) of the side lengths of the rectangles under consideration is infinity. As applications, we obtain the Hausdorff dimension of naturally occurring sets within the classical framework of simultaneous Diophantine approximation and the dynamical framework of shrinking target problems. For instance, concerning the former, for , let denote the set of simultaneously satisfying the inequalities and for infinitely many . Then, the `unbounded' Mass Transference Principle enables us to show that .

Paper Structure

This paper contains 18 sections, 15 theorems, 343 equations.

Table of Contents

  1. Introduction
  2. The Mass Transference Principle(s)
  3. Mass Transference Principle for rectangles
  4. Proof of Theorem \ref{['Runbounded']} modulo Theorem \ref{['pointcase']}
  5. The upper bound
  6. The lower bound modulo Theorem \ref{['pointcase']}
  7. Application to matrix transformation of tori
  8. Properties of the dimensional number
  9. Proof of Propositions \ref{['Prop:4']} & \ref{['ll1']}
  10. Proof of Theorem \ref{['pointcase']}
  11. Preliminaries for the proof
  12. Imposing additional assumptions and structure.
  13. The $K_{G,B}$ covering lemma for rectangles.
  14. Notions and sub-collections associated with the $K_{G,B}$ lemma.
  15. Constructing the Cantor set $\mathbb{E}_\infty(\epsilon)$ and the measure $\nu_\epsilon$
  16. ...and 3 more sections

Key Result

Theorem 1.1

For $1\leq i\leq d$, let $\psi_i: \mathbb{R}^+\to\mathbb{R}^+$ be a real, positive, non-increasing function. Then where given $\mathbf{t}=(t_1,\dots,t_d) \in ({\Bbb R}^+\cup \{+\infty\})^d$ we set $\mathcal{L}(\mathbf{t}):=\{1\leq i\leq d: t_i<+\infty\}$.

Theorems & Definitions (45)

  • Remark 1
  • Remark 2
  • Theorem 1.1
  • Remark 3
  • Corollary 1.1
  • Theorem 1.2
  • Remark 4
  • Remark 5
  • Remark 6
  • Theorem 2.1
  • ...and 35 more