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inf(M \ L)=3

Harold Erazo, Davi Lima, Carlos Matheus, Carlos Gustavo Moreira, Sandoel Vieira

Abstract

The Lagrange and Markov spectra $L$ and $M$ describe the best constants of Diophantine approximations for irrational numbers and binary quadratic forms. In 1880, A. Markov showed that the initial portions of these spectra coincide: indeed, $L\cap (0,3) = M\cap (0,3)$ is a discrete set of explicit quadratic irrationals accumulating only at $3$. In this article, we show that the statement above ceases to be true immediately after $3$: in particular, $L\cap (3,3+\varepsilon)\neq M\cap (3,3+\varepsilon)$ for all $\varepsilon>0$, and thus $\inf(M\setminus L)=3$. In fact, we derive this result as a by-product of lower bounds on the Hausdorff dimension of $(M\setminus L)\cap (3,3+\varepsilon)$ implying that $\liminf\limits_{\varepsilon\to 0} \frac{\dim_H((M\setminus L)\cap(3,3+\varepsilon))}{\dim_H(M\cap (3,3+\varepsilon))}\geq \frac{1}{2}$ and, as it turns out, these bounds are obtained from the study of projections of Cartesian products of almost affine dynamical Cantor sets via an argument of probabilistic flavor based on Baker--Wüstholz theorem on linear forms in logarithms of algebraic numbers.

inf(M \ L)=3

Abstract

The Lagrange and Markov spectra and describe the best constants of Diophantine approximations for irrational numbers and binary quadratic forms. In 1880, A. Markov showed that the initial portions of these spectra coincide: indeed, is a discrete set of explicit quadratic irrationals accumulating only at . In this article, we show that the statement above ceases to be true immediately after : in particular, for all , and thus . In fact, we derive this result as a by-product of lower bounds on the Hausdorff dimension of implying that and, as it turns out, these bounds are obtained from the study of projections of Cartesian products of almost affine dynamical Cantor sets via an argument of probabilistic flavor based on Baker--Wüstholz theorem on linear forms in logarithms of algebraic numbers.

Paper Structure

This paper contains 24 sections, 31 theorems, 250 equations, 5 figures.

Table of Contents

  1. Introduction
  2. Perron's description of the Markov and Lagrange spectra
  3. Fractal geometry of $M\setminus L$ close to $3$
  4. Outline of the proof of Theorem \ref{['thm:dimension']}
  5. Organization of the paper
  6. Preliminaries
  7. Some standard notation
  8. Some classical facts about continued fractions
  9. Intervals of the Gauss map
  10. Gauss--Cantor sets
  11. Hausdorff dimension of Gauss-Cantor sets with big blocks of 1's
  12. Linear forms in logarithms
  13. Estimating denominators of convergents
  14. Probabilistically unlikely combinatorics
  15. Projections of Cantor sets
  16. ...and 9 more sections

Key Result

Theorem 1.1

For any $\varepsilon>0$ we have Consequently, $\inf(M\setminus L)=3$.

Figures (5)

  • Figure 1: The tree of alphabets $\overline{T}$.
  • Figure 2: The product $K\times K$ projects into $K-K$ by the function $(x,y)\mapsto x-y$. Each rectangle $I(1^{e_1}221^{n_0})\times I(1^{e_{-1}}221^{n_0})$ has a disjoint projection from the others.
  • Figure 3: Markov partition of $K\times K$ projecting onto $K-K$.
  • Figure 4: No local uniqueness
  • Figure 5: Local uniqueness

Theorems & Definitions (58)

  • Theorem 1.1
  • Theorem 1.2
  • Remark 1.3
  • Lemma 2.1
  • Lemma 2.2
  • proof
  • Lemma 2.3: Bounded distortion
  • proof
  • Definition 2.4
  • Lemma 2.5
  • ...and 48 more