Hausdorff Dimension of Growth Rate Level Sets in $\theta$-expansions

1. Problem Statement and Context

The paper investigates the metric properties of $\theta$-expansions, a generalization of regular continued fractions. While classical limit theorems (e.g., Khinchine, Lévy) describe the "typical" behavior of digit sequences almost everywhere, they often fail to capture the structure of "exceptional" sets where digits grow at specific, non-typical rates.

The specific problem addressed is determining the Hausdorff dimension of the level sets defined by the relative growth rate between the largest digit and the sum of the remaining digits. Specifically, for a given $\alpha \ge 0$, the authors analyze the set:
$$E_\theta(\alpha) = \left\{ x \in [0, \theta] \setminus \mathbb{Q} : \lim_{n \to +\infty} \frac{L_{n,\theta}(x) \log n \log \log n}{S_{n,\theta}(x) - L_{n,\theta}(x)} = \alpha \right\}$$
where:

• $\theta \in (0, 1)$ is an irrational number such that $\theta^2 = 1/m$ for a non-square integer $m$.
• $L_{n,\theta}(x)$ is the maximum of the first $n$ digits.
• $S_{n,\theta}(x)$ is the sum of the first $n$ digits.

Goal: To prove that for any $\alpha \ge 0$, the set $E_\theta(\alpha)$ has full Hausdorff dimension (i.e., $\dim_H E_\theta(\alpha) = 1$), thereby extending a seminal result by Zhang and Lü (2016) from regular continued fractions ($\theta=1$) to the broader $\theta$-expansion framework.

2. Methodology

The proof relies on a constructive approach combined with fractal geometry and ergodic theory. The core strategy involves building explicit subsets of $E_\theta(\alpha)$ that are large enough to have dimension 1, while ensuring they map bijectively to sets of numbers with bounded digits (which are known to have full dimension).

Key Components of the Method:

1. Sparse Sequence Construction:
   Instead of using polynomial sequences common in previous works, the authors introduce a super-exponential sparse sequence to control the insertion of large digits:
   $$n_k = \lfloor \exp(k^{3/4}) \rfloor$$
   This sequence is chosen because the gaps $n_{k+1} - n_k$ are small relative to $n_k$ (specifically $o(n_k)$), yet the logarithmic growth of $n_k$ is slow enough to allow the inserted digits to dominate the limit ratio without disrupting the convergence of the sum.

2. Digit Insertion Strategy:
   The authors construct a subset $E_{M,\theta}(\alpha)$ by taking numbers with bounded digits (between $m$ and $M$) and inserting specific large digits at positions $n_k$. The inserted digit $\ell_{n_k}$ is defined as:
   $$\ell_{n_k}(x) = \left\lfloor \frac{\alpha \sum_{i=1}^{n_k-1} \ell_i(x)}{\log n_k \log \log n_k} \right\rfloor$$
   This ensures that the ratio in the definition of $E_\theta(\alpha)$ converges exactly to $\alpha$.

3. Dimension Preservation via Hölder Maps:
   To prove the constructed set has full dimension, the authors define a map $f: E_{M,\theta}(\alpha) \to X_M$, where $X_M$ is the set of numbers with digits uniformly bounded by $M$.

   • The Map: $f$ removes the inserted large digits at positions $n_k$, effectively "compressing" the expansion.
   • Hölder Continuity: The authors prove that $f$ and its inverse $f^{-1}$ are bi-Hölder continuous with an exponent arbitrarily close to 1 (specifically $(1-\varepsilon)$-Hölder).
   • Significance: Since bi-Hölder maps preserve Hausdorff dimension, and $X_M$ is known to have dimension approaching 1 as $M \to \infty$ (via a Jarník-type theorem), the constructed set $E_{M,\theta}(\alpha)$ must also have dimension 1.

4. Metric Estimates:
   The proof involves rigorous estimates of the lengths of cylinder intervals $I_n(\ell_1, \dots, \ell_n)$ in $\theta$-expansions. A critical step is bounding the distortion caused by the inserted large digits on the denominators of the convergents ($Q_n$), showing that the "cost" of inserting a large digit is negligible compared to the super-exponential growth of the denominators.

3. Key Contributions

• Generalization of Zhang and Lü (2016): The paper successfully extends the result regarding the full Hausdorff dimension of digit growth level sets from regular continued fractions to the general $\theta$-expansion framework.
• Novel Sparse Sequence: The introduction of the sequence $n_k = \lfloor \exp(k^{3/4}) \rfloor$ is a specific technical innovation tailored to the metric properties of $\theta$-expansions, differing from the polynomial sequences used in classical continued fraction studies.
• Refined Hölder Analysis: The paper provides a detailed verification of the Hölder continuity of the dimension-preserving map, specifically addressing the distortion introduced by the non-linear growth of digits in $\theta$-expansions.
• Unified Framework: It demonstrates that the interplay between ergodic theory (limit theorems) and fractal geometry (dimension of exceptional sets) holds robustly even when the underlying dynamical system is generalized beyond the standard Gauss map.

4. Main Results

Theorem 1.1:
Let $\theta \in (0, 1)$ be an irrational such that $\theta^2 = 1/m$ for a non-square $m \in \mathbb{N}^+$. For any $\alpha \ge 0$, the level set $E_\theta(\alpha)$ satisfies:
$$\dim_H E_\theta(\alpha) = 1$$

Implications:

• Case $\alpha = 0$: The set of numbers where the largest digit is negligible compared to the sum (bounded digits) already has full dimension.
• Case $\alpha > 0$: Even when the largest digit grows at a specific rate relative to the sum (controlled by $\alpha$), the set of such numbers is "large" in the sense of fractal dimension, despite being a null set with respect to the $\theta$-Gauss measure.

5. Significance

This work deepens the understanding of the metric theory of continued fractions and its generalizations.

• Richness of Exceptional Sets: It confirms that while limit theorems dictate the behavior of "almost all" numbers, the set of numbers exhibiting specific, atypical growth rates is topologically and dimensionally rich (full dimension).
• Robustness of Fractal Structure: The result suggests that the fractal geometry of digit growth is a universal feature of continued fraction-like expansions, regardless of the specific parameter $\theta$, provided the system retains certain ergodic and metric properties.
• Future Applications: The constructive method and the specific sparse sequence used here are adaptable to other growth rate problems in $\theta$-expansions, such as limits involving $\log \log n$ or different normalization factors, offering a template for future research in dynamical systems and number theory.