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On very badly approximable numbers

Zhe Cao, Harold Erazo, Carlos Gustavo Moreira

TL;DR

The paper delivers a complete combinatorial characterization of irrationals for which the inequality $\left|x-\frac{p}{q}\right|<\frac{1}{3q^2}$ has only finitely many solutions, tying the phenomenon to continued fractions whose tails are encoded by balanced Christoffel words via a renormalization framework. It shows that such numbers are either quadratic surds or transcendental, and derives a striking corollary: every algebraic real of degree at least $3$ admits infinitely many good rational approximations with the sharp constant $\frac{1}{3q^2}$. The work blends Diophantine approximation, dynamical systems, and combinatorics of words to produce an explicit, self-contained proof and a deep understanding of the initial segment of the Markov/tilde-M spectra, including precise links between $\mathcal{L}$, $\mathcal{M}$, and $\widetilde{\mathcal{M}}$. The results extend classical Markov theory with a renormalization viewpoint, yielding concrete criteria and constructive procedures for both periodic and nonperiodic continued fraction tails and clarifying the structure of the related spectra.

Abstract

We prove a refined version of Markov's theorem in Diophantine approximation. More precisely, we characterize completely the set of irrationals $x$ such that $\left|x-\frac{p}{q}\right|<\frac{1}{3q^2}$ has only finitely many rational solutions: their continued fraction is eventually a balanced sequence through a simple coding. As consequence, we show that all such numbers are either quadratic surds or transcendental numbers. In particular, for any algebraic real number $x$ of degree at least $3$ there are infinitely rational numbers $\frac{p}{q}$ such that $\left|x-\frac{p}{q}\right|<\frac{1}{3q^2}$.

On very badly approximable numbers

TL;DR

The paper delivers a complete combinatorial characterization of irrationals for which the inequality has only finitely many solutions, tying the phenomenon to continued fractions whose tails are encoded by balanced Christoffel words via a renormalization framework. It shows that such numbers are either quadratic surds or transcendental, and derives a striking corollary: every algebraic real of degree at least admits infinitely many good rational approximations with the sharp constant . The work blends Diophantine approximation, dynamical systems, and combinatorics of words to produce an explicit, self-contained proof and a deep understanding of the initial segment of the Markov/tilde-M spectra, including precise links between , , and . The results extend classical Markov theory with a renormalization viewpoint, yielding concrete criteria and constructive procedures for both periodic and nonperiodic continued fraction tails and clarifying the structure of the related spectra.

Abstract

We prove a refined version of Markov's theorem in Diophantine approximation. More precisely, we characterize completely the set of irrationals such that has only finitely many rational solutions: their continued fraction is eventually a balanced sequence through a simple coding. As consequence, we show that all such numbers are either quadratic surds or transcendental numbers. In particular, for any algebraic real number of degree at least there are infinitely rational numbers such that .
Paper Structure (28 sections, 47 theorems, 136 equations)

This paper contains 28 sections, 47 theorems, 136 equations.

Table of Contents

  1. Introduction
  2. Main Results
  3. Spectrum M-tilde
  4. Acknowledgments
  5. Preliminary
  6. Periodic continued fractions
  7. Markov's Theorem
  8. Badly approximable numbers
  9. Markov quadratic forms
  10. Dynamical Characterization
  11. Renormalization
  12. Some properties of Christoffel words
  13. Roots of Markov quadratic forms
  14. The spectrum Tilde-M before 3
  15. Cuts
  16. ...and 13 more sections

Key Result

Theorem 1.1

Let $x=[x_0;x_1,x_2,\dots]$ be such that $\left| x-\frac{p}{q} \right| < \frac{1}{3q^2}$ has only finitely many solutions and let $N\in\mathbb{N}$ minimal such that $\lambda_N(x)\leq3$ for all $n\geq N+1$. If the continued fraction of $x$ is ultimately periodic, then there is $(\alpha,\beta)\in\over Moreover, if $N\geq 1$ and $x_{N+1}x_{N+2}\dots\neq a^\infty, b^\infty$, then there exist unique fa

Theorems & Definitions (87)

  • Theorem 1.1
  • Remark 1.1
  • Remark 1.2
  • Remark 1.3
  • Theorem 1.2
  • Theorem 1.3
  • Theorem 1.4
  • Theorem 1.5
  • Theorem 2.1
  • Definition 2.1
  • ...and 77 more