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Generalized discrete Markov spectra

Yasuaki Gyoda

TL;DR

This work generalizes the discrete Markov spectrum by introducing the $(k_1,k_2,k_3)$-GM numbers from a generalized Markov equation and constructing the corresponding generalized discrete Markov spectrum $\mathcal{M}_{k_1,k_2,k_3}$. It leverages snake-graph methods and generalized Cohn matrices to derive a continued-fraction decomposition framework, yielding an explicit formula for the Markov-Lagrange spectrum values $\mathcal{L}([s(t)^{\infty}])=\frac{\sqrt{\Delta(n_t,i_t)}}{n_t}$ with $\Delta(n,i)=((3+k_1+k_2+k_3)n-k_i)^2-4$, and proving $\mathcal{M}_{k_1,k_2,k_3}\subset\mathcal{L}\subset\mathcal{M}$. The paper also establishes a transition-interval characterization of the union spectrum $\mathcal{M}'$ and discusses a generalized uniqueness conjecture, including counterexamples to naive extensions, highlighting the nuanced structure of GM spectra beyond the classical case. Overall, the work unifies combinatorial, geometric, and number-theoretic techniques to extend Markov-type phenomena to a broad family of Diophantine objects. The results provide new insights into the relationship between generalized Markov numbers, spectral theory, and continued-fraction/graph-theoretic representations with potential implications for Diophantine approximation and cluster-algebra-inspired methods.

Abstract

In this paper, we generalize the special subset of the Markov-Lagrange spectrum (and the Markov spectrum) called the discrete Markov spectrum. The discrete Markov spectrum is defined in terms of the Markov numbers, which arise as positive integer solutions to the Markov equation $x^2 + y^2 + z^2 = 3xyz.$ Using the tool called snake graphs, originating from cluster algebra theory, we first reconstruct proofs of its properties in a combinatorial framework and then extend it to the generalized setting. We then introduce the generalized discrete Markov spectrum, defined analogously via the generalized Markov numbers, which arise as positive integer solutions to the generalized Markov equation $x^2 + y^2 + z^2 + k_1 yz + k_2 zx + k_3 xy = (3 + k_1 + k_2 + k_3) xyz.$ We prove that this generalized spectrum is contained in the Markov-Lagrange spectrum and thus the Markov spectrum.

Generalized discrete Markov spectra

TL;DR

This work generalizes the discrete Markov spectrum by introducing the -GM numbers from a generalized Markov equation and constructing the corresponding generalized discrete Markov spectrum . It leverages snake-graph methods and generalized Cohn matrices to derive a continued-fraction decomposition framework, yielding an explicit formula for the Markov-Lagrange spectrum values with , and proving . The paper also establishes a transition-interval characterization of the union spectrum and discusses a generalized uniqueness conjecture, including counterexamples to naive extensions, highlighting the nuanced structure of GM spectra beyond the classical case. Overall, the work unifies combinatorial, geometric, and number-theoretic techniques to extend Markov-type phenomena to a broad family of Diophantine objects. The results provide new insights into the relationship between generalized Markov numbers, spectral theory, and continued-fraction/graph-theoretic representations with potential implications for Diophantine approximation and cluster-algebra-inspired methods.

Abstract

In this paper, we generalize the special subset of the Markov-Lagrange spectrum (and the Markov spectrum) called the discrete Markov spectrum. The discrete Markov spectrum is defined in terms of the Markov numbers, which arise as positive integer solutions to the Markov equation Using the tool called snake graphs, originating from cluster algebra theory, we first reconstruct proofs of its properties in a combinatorial framework and then extend it to the generalized setting. We then introduce the generalized discrete Markov spectrum, defined analogously via the generalized Markov numbers, which arise as positive integer solutions to the generalized Markov equation We prove that this generalized spectrum is contained in the Markov-Lagrange spectrum and thus the Markov spectrum.

Paper Structure

This paper contains 18 sections, 38 theorems, 164 equations, 14 figures, 12 tables.

Table of Contents

  1. Introduction and Main Results
  2. Background
  3. Main Results
  4. Organization
  5. Classical Discrete Markov Spectrum
  6. Markov Numbers and Markov Tree
  7. Cohn Words, Cohn Matrices, and Cohn Tree
  8. Continued Fraction Matrix Decompositions of Cohn Matrices
  9. Markov-Lagrange Spectrum and Markov Spectrum
  10. Generalized Markov Equation and Generalized Markov Numbers
  11. Generalized Cohn Matrices and their Decompositions
  12. Generalized Cohn Matrices
  13. Continued Fraction Matrix Decomposition of Generalized Cohn Matrices
  14. Generalized Markov Length and Generalized Markov Distance
  15. Generalized Discrete Markov Spectra
  16. ...and 3 more sections

Key Result

Theorem 1.1

Let Then $\mathcal{M}_d=\mathcal{M}_{< 3}$.

Figures (14)

  • Figure 1: The word $c_{\frac{2}{5}}$ and $c_{\frac{5}{2}}$
  • Figure 2: The line segment $L_t$ for $t=\frac{2}{5}$
  • Figure 3: The line segment $\overline{L_t}$ associated with $t=\frac{2}{5}$
  • Figure 4: Interpretation $\overline{L_t}$ as loop on torus for $t=\frac{1}{1}$
  • Figure 7: Signs of triangles intersecting $\overline{L_t}$ for $t=\frac{2}{5}$
  • ...and 9 more figures

Theorems & Definitions (90)

  • Theorem 1.1: Markov's theorem mar1mar2
  • Theorem 1.2: Theorem \ref{['thm:markov-value-gen']}, Corollaries \ref{['cor:markov-lagrange']}, \ref{['cor:inL']}
  • Theorem 1.4: Theorem \ref{['thm:0-2-relation']}
  • Theorem 1.5: Theorem \ref{['thm:freiman-interval']}
  • Conjecture 1.6: Conjecture \ref{['uniqueness-conjecture']}
  • Proposition 2.1: see gyoda-maruyama-sato*Proposition 3.2
  • Proposition 2.2: see aig*Corollary 3.4
  • Definition 2.3
  • Definition 2.4
  • Proposition 2.5: see aig*Section 3.2
  • ...and 80 more