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On the classical Lagrange and Markov spectra: new results on the local dimension and the geometry of the difference set

Harold Erazo, Luke Jeffreys, Carlos Gustavo Moreira

TL;DR

This work analyzes the fractal geometry of the classical Lagrange spectrum $L$ and Markov spectrum $M$, emphasizing local dimension via $d_{loc}$ and the structure of the difference set $M\setminus L$. By introducing and exploiting the concept of good intervals, the authors prove that $d_{loc}$ is non-decreasing on these regions and that $M'\cap I = M''\cap I$, with a complete description of the local dimension on $L'$, $M'$, and related sets within these intervals. They provide an explicit, computer-assisted construction of many good intervals, yielding precise endpoints defined by minimal Markov values of carefully chosen words and linking these to Cantor-set dynamics (Gauss-Cantor sets). The paper also identifies the largest known elements of $M\setminus L$ in several regions, and documents two new maximal gaps in $M$ near 3.942, accompanied by a detailed algorithmic framework and self-replication phenomena. Overall, the results sharpen the local and global understanding of $L$ and $M$ between Hall's ray and Freiman’s constant, and improve lower bounds on $\dim_H(M\setminus L)$ within good intervals.

Abstract

Let $L$ and $M$ denote the classical Lagrange and Markov spectra, respectively. It is known that $L\subset M$ and that $M\setminus L\neq\varnothing$. Inspired by three questions asked by the third author in previous work investigating the fractal geometric properties of the Lagrange and Markov spectra, we investigate the function $d_{loc}(t)$ that gives the local Hausdorff dimension at a point $t$ of $L'$. Specifically, we construct several intervals (having non-trivial intersection with $L'$) on which $d_{loc}$ is non-decreasing. We also prove that the respective intersections of $M'$ and $M''$ with these intervals coincide. Furthermore, we completely characterize the local dimension of both spectra when restricted to those intervals. Finally, we demonstrate the largest known elements of the difference set $M\setminus L$ and describe two new maximal gaps of $M$ nearby.

On the classical Lagrange and Markov spectra: new results on the local dimension and the geometry of the difference set

TL;DR

This work analyzes the fractal geometry of the classical Lagrange spectrum and Markov spectrum , emphasizing local dimension via and the structure of the difference set . By introducing and exploiting the concept of good intervals, the authors prove that is non-decreasing on these regions and that , with a complete description of the local dimension on , , and related sets within these intervals. They provide an explicit, computer-assisted construction of many good intervals, yielding precise endpoints defined by minimal Markov values of carefully chosen words and linking these to Cantor-set dynamics (Gauss-Cantor sets). The paper also identifies the largest known elements of in several regions, and documents two new maximal gaps in near 3.942, accompanied by a detailed algorithmic framework and self-replication phenomena. Overall, the results sharpen the local and global understanding of and between Hall's ray and Freiman’s constant, and improve lower bounds on within good intervals.

Abstract

Let and denote the classical Lagrange and Markov spectra, respectively. It is known that and that . Inspired by three questions asked by the third author in previous work investigating the fractal geometric properties of the Lagrange and Markov spectra, we investigate the function that gives the local Hausdorff dimension at a point of . Specifically, we construct several intervals (having non-trivial intersection with ) on which is non-decreasing. We also prove that the respective intersections of and with these intervals coincide. Furthermore, we completely characterize the local dimension of both spectra when restricted to those intervals. Finally, we demonstrate the largest known elements of the difference set and describe two new maximal gaps of nearby.
Paper Structure (49 sections, 66 theorems, 246 equations, 3 figures)

This paper contains 49 sections, 66 theorems, 246 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Continued fractions and shift space dynamics
  3. Local dimension
  4. Geometry of $\boldsymbol{M\setminus L}$
  5. $\boldsymbol{M'}$ and $\boldsymbol{M"}$
  6. Properties of $D(t)$
  7. Open questions
  8. Monotonicity of local dimension and good intervals
  9. Good intervals
  10. Local dimension on good intervals
  11. The topology and geometry of $\boldsymbol{M'}$ and the Hausdorff dimension of $M\setminus L$
  12. Good intervals and Berstein's investigation of $\boldsymbol{M\setminus L}$
  13. Properties of $D(t)$ on good intervals
  14. Construction of good intervals
  15. Interval [3.05082, 3.122183)
  16. ...and 34 more sections

Key Result

Theorem 1.1

Consider the intervals For each interval $I$ above, $I\cap L'\neq\varnothing$ and for all $t\in I\cap L^\prime$ we have In particular $d_{loc}|_{I\cap L'}$ is non-decreasing.

Figures (3)

  • Figure 1.1: The intervals of Theorem \ref{['thm:dloc']} are depicted in blue. Hall's ray is depicted in grey. The interval $[4.1,4.52]$ of Berstein's conjecture is depicted in red. The largest known value of $M\setminus L$, denoted in the figure as $m_1$, is given by \ref{['thm:M-L']}.
  • Figure 1.2: Structure of known regions of $M\setminus L$
  • Figure B.1: The good intervals produced in this paper before 4 are depicted in blue. Berstein's intervals are depicted in green.

Theorems & Definitions (134)

  • Theorem 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Theorem 1.4
  • Theorem 1.5
  • Proposition 1.6
  • Lemma 2.1
  • Lemma 2.2
  • Proposition 2.3
  • proof
  • ...and 124 more