On the graph of the dimension function of the Lagrange and Markov spectra

1. Problem Statement

The paper investigates the dimension function $d(t)$ associated with the classical Lagrange spectrum ($L$) and Markov spectrum ($M$). These spectra are subsets of the real line arising from the best constants in Diophantine approximations of irrational numbers and indefinite binary quadratic forms.

• Definitions:
  • $L := \{ \limsup_{n\to\infty} \lambda_n(\alpha) : \alpha \in (\mathbb{N}^*)^\mathbb{Z} \}$
  • $M := \{ \sup_{n\in\mathbb{Z}} \lambda_n(\alpha) : \alpha \in (\mathbb{N}^*)^\mathbb{Z} \}$
  • Where $\lambda_n(\alpha)$ relates to continued fraction expansions.
• The Function: The authors study $d(t) := \dim_H(L \cap (-\infty, t)) = \dim_H(M \cap (-\infty, t))$, where $\dim_H$ denotes the Hausdorff dimension.
• Key Properties: It is known that $L \subset M$ are closed sets. Consequently, their complements consist of countable unions of open intervals (gaps). On these gaps, $d(t)$ is constant, forming plateaux.
• Objective: The goal is to rigorously determine the graph of $d(t)$ between $3$ and the critical value $t_1 \approx 3.334384$ (where $d(t)$ reaches 1). Specifically, the authors aim to identify the twelve largest non-trivial plateaux and approximate the function's behavior between them.

2. Methodology

The authors employ a hybrid approach combining rigorous dynamical systems theory with high-precision numerical computation.

A. Theoretical Framework

1. Gauss-Cantor Sets: The dimension of the spectra is linked to the Hausdorff dimension of specific dynamical Cantor sets (Gauss-Cantor sets) defined by forbidden subsequences in continued fractions.
2. Lemma on Dimension Drop (Lemma 2.3): A crucial theoretical tool is the fact that removing an element from the Markov partition of a dynamical Cantor set strictly decreases its Hausdorff dimension. This allows the authors to prove that specific intervals are indeed plateaux (where the dimension is constant) by showing that extending the set of forbidden words reduces the dimension.
3. Lemma on Markov Values (Lemma 2.2): This lemma helps determine the minimum Markov value of sequences containing specific patterns, allowing the authors to pinpoint the exact endpoints of gaps in the spectrum.

B. Numerical Methods

1. Recursive Construction of Forbidden Words: To compute $d(t)$ at specific points, the authors use a recursive algorithm (Algorithm 1) to construct sets of "forbidden" finite strings.
   • For a given threshold $s$, the algorithm builds a tree of strings. If a string's continuations guarantee a Markov value $> s$, it is marked as forbidden.
   • This generates a Cantor set $K$ such that $M \cap (-\infty, s) \subset 2 + K + K$.
2. Bounding the Dimension:
   • Upper Bound: $d(s) \leq 2 \dim_H(K)$.
   • Lower Bound: By finding the maximum Markov value $s'$ of strings not containing forbidden substrings, they establish $d(s') \geq \min\{2 \dim_H(K), 1\}$.
3. Rigorous Verification: The numerical computations are performed with rigorous error bounds (using interval arithmetic concepts implicitly via the "checkpoints" and truncation of numbers) to ensure the results are mathematically valid, not just heuristic approximations.

3. Key Contributions

1. Identification of Plateaux: The paper explicitly identifies twelve non-trivial plateaux ($P_2$ through $P_{13}$) of the dimension function $d(t)$ in the interval $(3, t_1)$.
2. Exact Endpoints: For each plateau, the authors provide exact algebraic expressions (involving square roots) for the left and right endpoints, derived from the specific forbidden subsequences that define the gaps in the Markov spectrum.
3. Structural Insight: The authors reveal that the structure of these plateaux is complex. A single plateau may correspond to the union of multiple gaps in the Markov spectrum, sometimes separated by isolated points of the Lagrange spectrum.
   • Example: The largest plateau $P_2$ is the closure of the union of the largest gap $G_1$ and another gap $J_1$, where the connecting point is an isolated point of $L$.
4. Threshold Proof: They prove that the ten largest non-trivial plateaux are exactly those with lengths greater than 0.005.

4. Key Results

The main theorem (Theorem 3.1) states that the ten largest plateaux of $d(t)$ are precisely those with length $> 0.005$. The authors list these plateaux in decreasing order of length:

• Auxiliary Plateaux: Two additional plateaux ($P_{12}$ and $P_{13}$) were identified with lengths ~0.0042 and ~0.0040, respectively.
• Ordering: The plateaux do not appear in the order of their lengths on the real line. For instance, $P_2$ (the longest) appears after $P_3$ and $P_4$ in terms of $t$-values.
• Graph Approximation: The paper provides rigorous lower and upper bounds for $d(t)$ in the intervals between these plateaux, effectively sketching the "staircase" nature of the function.

5. Significance

• Resolution of a Long-standing Problem: The paper provides the first rigorous, detailed visualization of the dimension function $d(t)$ for the Lagrange and Markov spectra in the critical region between 3 and $t_1$.
• Methodological Advancement: It successfully bridges the gap between theoretical dynamical systems (Cantor sets, Markov partitions) and rigorous numerical computation. The use of "forbidden words" to bound dimensions is a powerful technique refined here.
• Understanding Fractal Structure: The work clarifies the intricate relationship between the gaps in the Markov spectrum and the plateaux of the dimension function. It demonstrates that the "steps" in the dimension function are not simple single gaps but often complex unions of gaps and isolated points.
• Data Availability: The authors provide the code used for these computations (hosted on GitHub), allowing for reproducibility and further exploration of the spectra.

In summary, this paper transforms the abstract understanding of the Lagrange and Markov spectra into a concrete, computable graph, proving that the "steps" in the dimension function are dominated by a specific set of twelve large plateaux, with the top ten being strictly larger than 0.005 in length.