On the structure of the dimension spectrum for continued fraction expansions

1. Problem Statement

The paper investigates the Hausdorff dimension spectrum of sets of continued fractions with coefficients restricted to specific infinite subsets of natural numbers $\mathbb{N}$.

Let $A \subseteq \mathbb{N}$ be an infinite set. The set of continued fractions with coefficients in $A$ is defined as:
$$J_A = \{x \in (0, 1) : x = [a_1, a_2, \dots] \text{ with } a_i \in A \text{ for all } i\}$$
The dimension spectrum of $A$, denoted $DS(A)$, is the set of all possible Hausdorff dimensions of subsets of $J_A$ generated by subsets of $A$:
$$DS(A) = \{ \dim_H(J_B) : B \subseteq A \}$$

The central question addressed is the topological structure of $DS(A)$. Specifically, the authors analyze whether $DS(A)$ is a full interval (i.e., $[0, \dim_H(J_A)]$) or if it contains gaps (intervals where no dimension exists) and regions where it is nowhere dense. This builds upon the "Texan Conjecture" (proven for $A=\mathbb{N}$), which states that $DS(\mathbb{N}) = [0, 1]$.

The paper focuses on three specific types of sets $A$:

1. Powers of integers: $P_q = \{q^n : n \in \mathbb{N}\}$ for $q \ge 2$.
2. Powers including 1: $P^*_q = P_q \cup \{1\}$.
3. Monomials: $M_q = \{n^q : n \in \mathbb{N}\}$ for $q \in \mathbb{N}$.

2. Methodology

The authors employ a combination of spectral theory of Perron-Frobenius (transfer) operators and rigorous numerical estimation.

A. Perron-Frobenius Operators

For a finite set $F \subset \mathbb{N}$ and a parameter $s \ge 0$, the transfer operator $L_{s,F}$ acts on the Banach space of continuous functions $C([0,1])$:
$$(L_{s,F}f)(x) = \sum_{n \in F} \left(\frac{1}{n+x}\right)^{2s} f\left(\frac{1}{n+x}\right)$$
Key theoretical tools used include:

• Spectral Radius and Dimension: By Theorem 1.6, the Hausdorff dimension $\dim_H(J_F)$ is the unique $s$ such that the spectral radius $r(L_{s,F}) = 1$.
• Strict Positive Eigenvectors: The existence of a unique strictly positive eigenvector $v_{s,F}$ allows for precise bounds on the spectral radius.
• Monotonicity: The function $s \mapsto r(L_{s,F})$ is strictly decreasing.

B. "Strict Break Points" and Gap Analysis

The core analytical technique involves the concept of strict break points.

• A point $a_k \in A$ is a "break point" for a finite set $F$ and dimension $s$ if adding $a_k$ to $F$ pushes the dimension above $s$.
• Lemma 3.2 & 3.3: These lemmas establish that if a set $A$ allows for the construction of nested finite subsets where "break points" can be found arbitrarily far out in the sequence, then the dimension spectrum is full (contains the interval).
• Gap Identification: Conversely, if removing a specific element (or tail) from a set results in a dimension strictly lower than adding a specific small element, a gap is created in the spectrum. The authors use Lemma 3.4 to prove that if certain inequalities hold between dimensions of sets with and without specific elements, the spectrum is nowhere dense in the resulting interval.

C. Rigorous Numerical Bounds

To handle specific cases (particularly for $M_q$), the authors use numerical methods developed by Falk and Nussbaum. They compute rigorous upper and lower bounds for $\dim_H(J_F)$ for finite sets $F$ using the transfer operator and interval arithmetic (implemented in MATLAB). These bounds are critical for proving the existence of gaps for specific values of $q$.

3. Key Contributions and Results

A. Full Dimension Spectrum for Powers ($P_q$)

• Theorem 1.1: The authors prove a general criterion: If $A = \{a_n\}$ satisfies $a_n a_m \ge a_{n+m}$ for all $m \ge n \ge 1$ (with $m \ge 2$), then $DS(A) = [0, \dim_H(J_A)]$.
• Corollary 1.2: This implies that the set of powers $P_q = \{q^n : n \in \mathbb{N}\}$ has a full dimension spectrum for all $q \ge 2$. This answers a question posed by Chousionis, Leykekhman, and Urbański.
• Contrast with $P^*_q$: The set $P^*_q = \{1, q, q^2, \dots\}$ behaves differently.
  • Theorem 1.3: For $q \ge 3$, $DS(P^*_q)$ has many gaps. Specifically, there exist intervals $(\mu_k, \nu_k)$ where the spectrum is empty, and regions where it is nowhere dense.
  • Theorem 1.4: Despite the gaps, $DS(P^*_q)$ always contains an initial interval $[0, \frac{\ln 2}{2 \ln q}]$.

B. Structure of Monomial Sets ($M_q$)

The paper provides a complete classification of the dimension spectrum for $M_q = \{n^q : n \in \mathbb{N}\}$:

• Full Spectrum for Small $q$: For $q \in \{1, 2, 3, 4, 5\}$, $DS(M_q)$ is a full interval $[0, \dim_H(J_{M_q})]$.
• Gaps for $q \ge 6$:
  • Theorem 1.5(ii): For $q \ge 6$, a gap appears between $\dim_H(J_{M_q \setminus \{2^q\}})$ and $\dim_H(J_{\{1, 2^q\}})$.
  • Specific Structures:
    • For $q \in \{6, 7, 8\}$, $DS(M_q)$ is the disjoint union of two nontrivial closed intervals.
    • For $q \in \{9, 10\}$, $DS(M_q)$ is the disjoint union of three nontrivial closed intervals.
    • For $q \ge 11$, $DS(M_q)$ is the disjoint union of finitely many nontrivial closed intervals.
• Critical Break Point Value: The authors introduce the concept of a "critical break point value" $k^*$. They prove that if a set $A$ has such a value (which $M_q$ does for $q \ge 11$), its dimension spectrum must be a finite union of disjoint closed intervals (Theorem 8.3).

4. Significance

1. Resolution of Open Questions: The paper definitively answers the question regarding the fullness of the dimension spectrum for integer powers $P_q$, confirming it is full, while simultaneously showing that adding the element '1' ($P^*_q$) destroys this property, creating a complex structure of gaps.
2. New Phenomena in Fractal Geometry: The results on $M_q$ demonstrate that dimension spectra can be finite unions of disjoint intervals. This challenges the intuition that such spectra are typically single intervals or have infinite complexity. It provides concrete examples of conformal iterated function systems (CIFS) with a prescribed finite number of connected components in their dimension spectrum.
3. Methodological Advancement: The paper successfully integrates rigorous spectral theory (Perron-Frobenius operators) with computer-assisted proofs. The use of strict break points and the "critical break point value" provides a robust framework for analyzing the topological structure of dimension spectra for other infinite subsets of $\mathbb{N}$.
4. Open Problems: The authors identify that while the number of intervals in $DS(M_q)$ increases with $q$ (1 interval for $q \le 5$, 2 for $6 \le q \le 8$, 3 for $9 \le q \le 10$), it remains unknown if this number grows indefinitely or if there is an upper bound. They also pose questions about the "nowhere dense" nature of the gaps in $P^*_q$.

In summary, this work significantly advances the understanding of the fractal geometry of continued fractions by characterizing the dimension spectrum for a wide class of coefficient sets, revealing a rich interplay between the arithmetic properties of the set $A$ and the topological structure of its dimension spectrum.