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Hausdorff Dimension of Growth Rate Level Sets in $θ$-expansions

Andreas Rusu, Gabriela Ileana Sebe

Abstract

We investigate the Hausdorff dimension of level sets defined by digit growth rates in $θ$-expansions, a generalization of regular continued fractions. For any $α\geq 0$, we prove that the set \[ E_θ(α) = \left\{ x \in [0, θ] \setminus \mathbb{Q} : \lim_{n \to {+}\infty} \frac{L_{n,θ}(x) \log n \log \log n}{S_{n,θ}(x) - L_{n,θ}(x)} = α\right\} \] has full Hausdorff dimension. This extends previous work of Zhang and {Lü} (2016) on regular continued fractions to the broader framework of $θ$-expansions. The proof involves constructing explicit subsets with controlled digit growth and establishing dimension preservation through Hölder-continuous mappings.

Hausdorff Dimension of Growth Rate Level Sets in $θ$-expansions

Abstract

We investigate the Hausdorff dimension of level sets defined by digit growth rates in -expansions, a generalization of regular continued fractions. For any , we prove that the set \[ E_θ(α) = \left\{ x \in [0, θ] \setminus \mathbb{Q} : \lim_{n \to {+}\infty} \frac{L_{n,θ}(x) \log n \log \log n}{S_{n,θ}(x) - L_{n,θ}(x)} = α\right\} \] has full Hausdorff dimension. This extends previous work of Zhang and {Lü} (2016) on regular continued fractions to the broader framework of -expansions. The proof involves constructing explicit subsets with controlled digit growth and establishing dimension preservation through Hölder-continuous mappings.

Paper Structure

This paper contains 14 sections, 10 theorems, 92 equations.

Table of Contents

  1. Introduction
  2. Preliminaries on $\theta$-Expansions
  3. Dynamical System and Digit Sequences
  4. Ergodic and metric properties
  5. A Jarník-Type Theorem and Dimension Preservation
  6. Proof of Theorem \ref{['thm:main']}
  7. The Case $\alpha = 0$: Bounded Digits Suffice
  8. The Case $\alpha > 0$ and Construction of the Sparse Sequence
  9. Choice of the Sparse Sequence
  10. Technical Conditions and Set Construction
  11. Verification of the Limit Behavior
  12. Dimension Preservation via Hölder-Continuous Map
  13. Completion of the Proof
  14. Conclusion

Key Result

Theorem 1.1

Let $\theta\in(0,1)$ be an irrational such that $\theta^2=1/m$ for some non‑square $m\in\mathbb{N}_+$. For any $\alpha \geq 0$, the set satisfies $\dim_H E_\theta(\alpha) = 1$. $\blacktriangleleft$$\blacktriangleleft$

Theorems & Definitions (24)

  • Theorem 1.1
  • Definition 2.1: Generalized Gauss Map
  • Definition 2.2: $\theta$-Gauss Measure
  • Proposition 2.3: Metric Properties
  • Theorem 2.4: Jarník-type result for $\theta$-expansions
  • Lemma 2.5: Hölder Maps and Dimension
  • Lemma 3.1
  • proof
  • Lemma 3.2: Properties of the Sparse Sequence
  • proof
  • ...and 14 more