The Lagrange and Markov spectra from the dynamical point of view

TL;DR

This paper explores the structure of the Lagrange and Markov spectra from a dynamical systems perspective, emphasizing recent developments and theorems in the field, particularly those by C. G. Moreira.

Contribution

It provides a dynamical viewpoint on the Lagrange and Markov spectra, highlighting recent theorems and structural insights in the context of ergodic theory and number theory.

Findings

Analysis of the structure of Lagrange and Markov spectra

Discussion of recent theorems by C. G. Moreira

Connection between spectra and dynamical systems

Abstract

This text grew out of my lecture notes for a 4-hours minicourse delivered on October 17 \& 19, 2016 during the research school "Applications of Ergodic Theory in Number Theory" -- an activity related to the Jean-Molet Chair project of Mariusz Lema\'nczyk and S\'ebastien Ferenczi -- realized at CIRM, Marseille, France. The subject of this text is the same of my minicourse, namely, the structure of the so-called Lagrange and Markov spectra (with an special emphasis on a recent theorem of C. G. Moreira).

Equations

$$\left|\alpha -\frac{p}{q}\right|\le \frac{1}{2q}$$

$$\left|\alpha -\frac{p}{q}\right|\le \frac{1}{q^2}$$

$$[0,1)=\bigcup _{j=0}^{Q-1}\left[\frac{j}{Q},\frac{j+1}{Q}\right)$$

$$|\{m\alpha \}-\{n\alpha \}|<\frac{1}{Q},$$

$$\left|\alpha -\frac{p}{q}\right|<\frac{1}{qQ}\le \frac{1}{q^2}$$

$$\left|\alpha -\frac{p}{q}\right|\le \frac{1}{\sqrt{5}{q}^2}$$

$$\left|\frac{1+\sqrt{5}}{2}-\frac{p}{q}\right|\le \frac{1}{(\sqrt{5}+\epsilon )q^2}$$

$$\#\left\{\frac{p}{q}\in \mathbb{Q}:\left|\frac{1+\sqrt{5}}{2}-\frac{p}{q}\right|\le \frac{1}{(\sqrt{5}+\epsilon )q^2}\right\}$$

$$\ell (\alpha ):=\underset{p,q\to \infty }{\mathrm{limsup}}\frac{1}{|q(q\alpha -p)|}$$

$$L:=\{\ell (\alpha ):\alpha \in \mathbb{R}-\mathbb{Q},\ell (\alpha )<\infty \}\subset \mathbb{R}$$

$$M:=\left\{\frac{\sqrt{\Delta (q)}}{\underset{(x,y)\in {\mathbb{Z}}^2-\{(0,0)\}}{\inf }|q(x,y)|}\in \mathbb{R}:q\text{ is an indefinite binary quadratic form with }\Delta (q)>0\right\}$$

$$V_1(x,y,z)=(x^{\prime },y,z)$$

$$\ell (\alpha )=\underset{n\to \infty }{\mathrm{limsup}}\frac{1}{|s_n(s_n\alpha -r_n)|}$$

$$\left|\alpha -\frac{p}{q}\right|<\frac{1}{2q^2}$$

$$\alpha =a_0+\frac{1}{a_1+\frac{1}{a_2+\frac{1}{\ddots }}}=:[a_0;a_1,a_2,\dots ]$$

$$\mathbb{Q}\ni \frac{p_n}{q_n}:=a_0+\frac{1}{a_1+\frac{1}{\ddots +\frac{1}{a_n}}}:=[a_0;a_1,\dots ,a_n]$$

$$\left\{\begin{array}[]{cc}p_{n+2}=a_{n+2}p_{n+1}+p_{n},&p_{-1}=1,p_{-2}=0\\ q_{n+2}=a_{n+2}q_{n+1}+q_{n},&q_{-1}=0,q_{-2}=1\end{array}\right.$$

$$[a_0;a_1,\dots ,a_{n-1},z]=\frac{z{p}_{n-1}+p_{n-2}}{z{q}_{n-1}+q_{n-2}}$$

$$\left(\begin{array}[]{cc}p_{n+1}&p_{n}\\ q_{n+1}&q_{n}\end{array}\right)\cdot\left(\begin{array}[]{cc}a_{n+2}&1\\ 1&0\end{array}\right)=\left(\begin{array}[]{cc}p_{n+2}&p_{n+1}\\ q_{n+2}&q_{n+1}\end{array}\right)$$

$$\text{either }\left|\alpha -\frac{p_n}{q_n}\right|<\frac{1}{2q_n^2}\text{ or }\left|\alpha -\frac{p_{n+1}}{q_{n+1}}\right|<\frac{1}{2q_{n+1}^2}.$$

$$\left|\frac{p_{n+1}}{q_{n+1}}-\frac{p_n}{q_n}\right|=\left|\frac{p_{n+1}{q}_n-p_n{q}_{n+1}}{q_n{q}_{n+1}}\right|=\left|\frac{(-1{)}^n}{q_n{q}_{n+1}}\right|=\frac{1}{q_n{q}_{n+1}}$$

$$\left|\alpha -\frac{p_n}{q_n}\right|\ge \frac{1}{2q_n^2}\text{ and }\left|\alpha -\frac{p_{n+1}}{q_{n+1}}\right|\ge \frac{1}{2q_{n+1}^2},$$

$$\frac{1}{q_n{q}_{n+1}}\ge \frac{1}{2q_n^2}+\frac{1}{2q_{n+1}^2},$$

$$\left|\frac{p}{q}-\frac{p_n}{q_n}\right|\ge \frac{1}{q{q}_n}>\frac{1}{q_n{q}_{n+1}}=\left|\frac{p_{n+1}}{q_{n+1}}-\frac{p_n}{q_n}\right|$$

$$\left|\alpha -\frac{p_n}{q_n}\right|<\left|\alpha -\frac{p}{q}\right|$$

$$\frac{p_0}{q_0}=3,\frac{p_1}{q_1}=\frac{22}{7},\frac{p_2}{q_2}=\frac{333}{106},\frac{p_3}{q_3}=\frac{355}{113},\dots$$

$$\left|\pi -\frac{22}{7}\right|<\frac{1}{700}<\left|\pi -\frac{314}{100}\right|\text{ and }\left|\pi -\frac{355}{113}\right|<\frac{1}{3,000,000}<\left|\pi -\frac{3141592}{1,000,000}\right|$$

$$\ell (\alpha )=\underset{n\to +\infty }{\mathrm{limsup}}f({\sigma }^n(\underline{\theta }))$$

$$L=\{\ell (\underline{\theta }):\underline{\theta }\in \Sigma ,\ell (\underline{\theta })<\infty \}$$

$$M=\{m(\underline{\theta }):\underline{\theta }\in \Sigma ,m(\underline{\theta })<\infty \}$$

Full text

1. 1 Diophantine approximations & Lagrange and Markov spectra

2. 1.1 Rational approximations of real numbers

3. 1.2 Integral values of binary quadratic forms

4. 1.3 Best rational approximations and continued fractions

5. 1.4 Perron’s characterization of Lagrange and Markov spectra

6. 1.5 Digression: Lagrange spectrum and cusp excursions on the modular surface

7. 1.6 Hall’s ray and Freiman’s constant

8. 1.7 Statement of Moreira’s theorem

9. 1.8 Hausdorff dimension

10. 2 Proof of Moreira’s theorem

11. 2.1 Strategy of proof of Moreira’s theorem

12. 2.2 Dynamical Cantor sets

13. 2.3 Gauss-Cantor sets

14. 2.4 Non-essentially affine Cantor sets

15. 2.5 Moreira’s dimension formula

16. 2.6 First step towards Moreira’s theorem 37: projections of Gauss-Cantor sets

17. 2.7 Second step towards Moreira’s theorem 37: upper semi-continuity

18. 2.8 Third step towards Moreira’s theorem 37: lower semi-continuity

19. 2.9 End of proof of Moreira’s theorem 37

20. A Proof of Hurwitz theorem

21. B Proof of Euler’s remark

1. Diophantine approximations & Lagrange and Markov spectra

1.1. Rational approximations of real numbers

Given a real number $\alpha\in\mathbb{R}$, it is natural to compare the quality $|\alpha-p/q|$ of a rational approximation $p/q\in\mathbb{Q}$ and the size $q$ of its denominator.

Since any real number lies between two consecutive integers, for every $\alpha\in\mathbb{R}$ and $q\in\mathbb{N}$, there exists $p\in\mathbb{Z}$ such that $|q\alpha-p|\leq 1/2$, i.e.

[TABLE]

In 1842, Dirichlet [4] used his famous pigeonhole principle to improve (1.1).

Theorem 1** (Dirichlet).**

For any $\alpha\in\mathbb{R}-\mathbb{Q}$, the inequality

[TABLE]

has infinitely many rational solutions $p/q\in\mathbb{Q}$.

Proof.

Given $Q\in\mathbb{N}$, we decompose the interval $[0,1)$ into $Q$ disjoint subintervals as follows:

[TABLE]

Next, we consider the $Q+1$ distinct111$\alpha\notin\mathbb{Q}$ is used here numbers $\{i\alpha\}$, $i=0,\dots,Q$, where $\{x\}$ denotes the fractional part222$\{x\}:=x-\lfloor x\rfloor$ and $\lfloor x\rfloor:=\max\{n\in\mathbb{Z}:n\leq x\}$ is the integer part of $x$. of $x$. By the pigeonhole principle, some interval $\left[\frac{j}{Q},\frac{j+1}{Q}\right)$ must contain two such numbers, say $\{n\alpha\}$ and $\{m\alpha\}$, $0\leq n<m\leq Q$. It follows that

[TABLE]

i.e., $|q\alpha-p|<1/Q$ where $0<q:=m-n\leq Q$ and $p:=\lfloor m\alpha\rfloor-\lfloor n\alpha\rfloor$. Therefore,

[TABLE]

This completes the proof of the theorem. ∎

In 1891, Hurwitz [12] showed that Dirichlet’s theorem is essentially optimal:

Theorem 2** (Hurwitz).**

For any $\alpha\in\mathbb{R}-\mathbb{Q}$, the inequality

[TABLE]

has infinitely many rational solutions $p/q\in\mathbb{Q}$.

Moreover, for all $\varepsilon>0$, the inequality

[TABLE]

has only finitely many rational solutions $p/q\in\mathbb{Q}$.

The first part of Hurwitz theorem is proved in Appendix A, while the second part of Hurwitz theorem is left as an exercise to the reader:

Exercise 3**.**

Show the second part of Hurwitz theorem. (Hint: use the identity $p^{2}-pq-q^{2}=\left(q\frac{1+\sqrt{5}}{2}-p\right)\left(q\frac{1-\sqrt{5}}{2}-p\right)$ relating $\frac{1+\sqrt{5}}{2}$ and its Galois conjugate $\frac{1-\sqrt{5}}{2}$).

Moreover, use your argument to give a bound on

[TABLE]

in terms of $\varepsilon>0$.

Note that Hurwitz theorem does not forbid an improvement of “$\left|\alpha-\frac{p}{q}\right|\leq\frac{1}{\sqrt{5}q^{2}}$ has infinitely many rational solutions $p/q\in\mathbb{Q}$” for certain $\alpha\in\mathbb{R}-\mathbb{Q}$. This motivates the following definition:

Definition 4**.**

The constant

[TABLE]

is called the best constant of Diophantine approximation of $\alpha$.

Intuitively, $\ell(\alpha)$ is the best constant $\ell$ such that $|\alpha-\frac{p}{q}|\leq\frac{1}{\ell q^{2}}$ has infinitely many rational solutions $p/q\in\mathbb{Q}$.

Remark 5*.*

By Hurwitz theorem, $\ell(\alpha)\geq\sqrt{5}$ for all $\alpha\in\mathbb{R}-\mathbb{Q}$ and $\ell(\frac{1+\sqrt{5}}{2})=\sqrt{5}$.

The collection of finite best constants of Diophantine approximations is the Lagrange spectrum:

Definition 6**.**

The Lagrange spectrum is

[TABLE]

Remark 7*.*

Khinchin proved in 1926 a famous theorem implying that $\ell(\alpha)=\infty$ for Lebesgue almost every $\alpha\in\mathbb{R}-\mathbb{Q}$ (see, e.g., Khinchin’s book [15] for more details).

1.2. Integral values of binary quadratic forms

Let $q(x,y)=ax^{2}+bxy+cy^{2}$ be a binary quadratic form with real coefficients $a,b,c\in\mathbb{R}$. Suppose that $q$ is indefinite333I.e., $q$ takes both positive and negative values. with positive discriminant $\Delta(q):=b^{2}-4ac$. What is the smallest value of $q(x,y)$ at non-trivial integral vectors $(x,y)\in\mathbb{Z}^{2}-\{(0,0)\}$?

Definition 8**.**

The Markov spectrum is

[TABLE]

Remark 9*.*

A similar Diophantine problem for ternary (and $n$-ary, $n\geq 3$) quadratic forms was proposed by Oppenheim in 1929. Oppenheim’s conjecture was famously solved in 1987 by Margulis using dynamics on homogeneous spaces: the reader is invited to consult Witte Morris book [28] for more details about this beautiful portion of Mathematics.

In 1880, Markov [17] noticed a relationship between certain binary quadratic forms and rational approximations of certain irrational numbers. This allowed him to prove the following result:

Theorem 10** (Markov).**

$L\cap(-\infty,3)=M\cap(-\infty,3)=\{k_{1}<k_{2}<k_{3}<k_{4}<\dots\}$* where $k_{1}=\sqrt{5}$, $k_{2}=\sqrt{8}$, $k_{3}=\frac{\sqrt{221}}{5}$, $k_{4}=\frac{\sqrt{1517}}{13}$, $\dots$ is an explicit increasing sequence of quadratic surds444I.e., $k_{n}^{2}\in\mathbb{Q}$ for all $n\in\mathbb{N}$. accumulating at $3$.*

In fact, $k_{n}=\sqrt{9-\frac{4}{m_{n}^{2}}}$ where $m_{n}\in\mathbb{N}$ is the $n$-th Markov number, and a Markov number is the largest coordinate of a Markov triple $(x,y,z)$, i.e., an integral solution of $x^{2}+y^{2}+z^{2}=3xyz$.

Remark 11*.*

All Markov triples can be deduced from $(1,1,1)$ by applying the so-called Vieta involutions $V_{1},V_{2},V_{3}$ given by

[TABLE]

where $x^{\prime}=3yz-x$ is the other solution of the second degree equation $X^{2}-3yzX+(y^{2}+z^{2})=0$, etc. In other terms, all Markov triples appear in Markov tree555Namely, the tree where Markov triples $(x,y,z)$ are displayed after applying permutations to put them in normalized form $x\leq y\leq z$, and two normalized Markov triples are connected if we can obtain one from the other by applying Vieta involutions.:

Remark 12*.*

For more informations on Markov numbers, the reader might consult Zagier’s paper [29] on this subject. Among many conjectures and results mentioned in this paper, we have:

• •

Conjecturally, each Markov number $z$ determines uniquely Markov triples $(x,y,z)$ with $x\leq y\leq z$;

• •

If $M(x):=\#\{m\textrm{ Markov number}:m\leq x\}$, then $M(x)=c(\log x)^{2}+O(\log x(\log\log x)^{2})$ for an explicit constant $c\simeq 0.18071704711507...$; conjecturally, $M(x)=c(\log(3x))^{2}+o(\log x)$, i.e., if $m_{n}$ is the $n$-th Markov number (counted with multiplicity), then $m_{n}\sim\frac{1}{3}A^{\sqrt{n}}$ with $A=e^{1/\sqrt{c}}\simeq 10.5101504...$

1.3. Best rational approximations and continued fractions

The constant $\ell(\alpha)$ was defined in terms of rational approximations of $\alpha\in\mathbb{R}-\mathbb{Q}$. In particular,

[TABLE]

where $(r_{n}/s_{n})_{n\in\mathbb{N}}$ is the sequence of best rational approximations of $\alpha$. Here, $p/q$ is called a best rational approximation666This nomenclature will be justified later by Propositions 18 and 19 below. whenever

[TABLE]

The sequence $(r_{n}/s_{n})_{n\in\mathbb{N}}$ of best rational approximations of $\alpha$ is produced by the so-called continued fraction algorithm.

Given $\alpha=\alpha_{0}\notin\mathbb{Q}$, we define recursively $a_{n}=\lfloor\alpha_{n}\rfloor$ and $\alpha_{n+1}=\frac{1}{\alpha_{n}-a_{n}}$ for all $n\in\mathbb{N}$. We can write $\alpha$ as a continued fraction

[TABLE]

and we denote

[TABLE]

Remark 13*.*

Lévy’s theorem [16] (from 1936) says that $\sqrt[n]{q_{n}}\to e^{\pi^{2}/12\log 2}\simeq 3.27582291872...$ for Lebesgue almost every $\alpha\in\mathbb{R}$. By elementary properties of continued fractions (recalled below), it follows from Lévy’s theorem that $\sqrt[n]{|\alpha-\frac{p_{n}}{q_{n}}|}\to e^{-\pi^{2}/6\log 2}\simeq 0.093187822954...$ for Lebesgue almost every $\alpha\in\mathbb{R}$.

Proposition 14**.**

$p_{n}$* and $q_{n}$ are recursively given by*

[TABLE]

Proof.

Exercise777Hint: Use induction and the fact that $[t_{0};t_{1},\dots,t_{n},t_{n+1}]=[t_{0};t_{1},\dots,t_{n}+\frac{1}{t_{n+1}}]$.. ∎

In other words, we have

[TABLE]

or, equivalently,

[TABLE]

Corollary 15**.**

$p_{n+1}q_{n}-p_{n}q_{n+1}=(-1)^{n}$* for all $n\geq 0$.*

Proof.

This follows from (1.3) because the matrix \left(\begin{array}[]{cc}\ast&1\\ 1&0\end{array}\right) has determinant $-1$. ∎

Corollary 16**.**

$\alpha=\frac{\alpha_{n}p_{n-1}+p_{n-2}}{\alpha_{n}q_{n-1}+q_{n-2}}$* and $\alpha_{n}=\frac{p_{n-2}-q_{n-2}\alpha}{q_{n-1}\alpha-p_{n-1}}$.*

Proof.

This is a consequence of (1.2) and the fact that $\alpha=:[a_{0};a_{1},\dots,a_{n-1},\alpha_{n}]$. ∎

The relationship between $\frac{p_{n}}{q_{n}}$ and the sequence of best rational approximations is explained by the following two propositions:

Proposition 17**.**

$\left|\alpha-\frac{p_{n}}{q_{n}}\right|\leq\frac{1}{q_{n}q_{n+1}}<\frac{1}{a_{n+1}q_{n}^{2}}\leq\frac{1}{q_{n}^{2}}$* and, moreover, for all $n\in\mathbb{N}$,*

[TABLE]

Proof.

Note that $\alpha$ belongs to the interval with extremities $p_{n}/q_{n}$ and $p_{n+1}/q_{n+1}$ (by Corollary 16). Since this interval has size

[TABLE]

(by Corollary 15), we conclude that $|\alpha-\frac{p_{n}}{q_{n}}|\leq\frac{1}{q_{n}q_{n+1}}$.

Furthermore, $\frac{1}{q_{n}q_{n+1}}=|\frac{p_{n+1}}{q_{n+1}}-\alpha|+|\alpha-\frac{p_{n}}{q_{n}}|$. Thus, if

[TABLE]

then

[TABLE]

i.e., $2q_{n}q_{n+1}\geq q_{n}^{2}+q_{n+1}^{2}$, i.e., $q_{n}=q_{n+1}$, a contradiction. ∎

In other terms, the sequence $(p_{n}/q_{n})_{n\in\mathbb{N}}$ produced by the continued fraction algorithm contains best rational approximations with frequency at least $1/2$.

Conversely, the continued fraction algorithm detects all best rational approximations:

Proposition 18**.**

If $|\alpha-\frac{p}{q}|<\frac{1}{2q^{2}}$, then $p/q=p_{n}/q_{n}$ for some $n\in\mathbb{N}$.

Proof.

Exercise888Hint: Take $q_{n-1}<q\leq q_{n}$, suppose that $p/q\neq p_{n}/q_{n}$ and derive a contradiction in each case $q=q_{n}$, $q_{n}/2\leq q<q_{n}$ and $q<q_{n}/2$ by analysing $|\alpha-\frac{p}{q}|$ and $|\frac{p}{q}-\frac{p_{n}}{q_{n}}|$ like in the proof of Proposition 19.. ∎

The terminology “best rational approximation” is motivated by the previous proposition and the following result:

Proposition 19**.**

For all $q<q_{n}$, we have $|\alpha-\frac{p_{n}}{q_{n}}|<|\alpha-\frac{p}{q}|$.

Proof.

If $q<q_{n+1}$ and $p/q\neq p_{n}/q_{n}$, then

[TABLE]

Hence, $p/q$ does not belong to the interval with extremities $p_{n}/q_{n}$ and $p_{n+1}/q_{n+1}$, and so

[TABLE]

because $\alpha$ lies between $p_{n}/q_{n}$ and $p_{n+1}/q_{n+1}$. ∎

In fact, the approximations $(p_{n}/q_{n})$ of $\alpha$ are usually quite impressive:

Example 20**.**

$\pi=[3;7,15,1,292,1,1,1,2,1,3,1,14,2,1,\dots]$* so that*

[TABLE]

The approximations $p_{1}/q_{1}$ and $p_{3}/q_{3}$ are called Yuelü and Milü (after Wikipedia) and they are somewhat spectacular:

[TABLE]

1.4. Perron’s characterization of Lagrange and Markov spectra

In 1921, Perron interpreted $\ell(\alpha)$ in terms of Dynamical Systems as follows.

Proposition 21**.**

$\alpha-\frac{p_{n}}{q_{n}}=\frac{(-1)^{n}}{(\alpha_{n+1}+\beta_{n+1})q_{n}^{2}}$* where $\beta_{n+1}:=\frac{q_{n-1}}{q_{n}}=[0;a_{n},a_{n-1},\dots,a_{1}]$.*

Proof.

Recall that $\alpha_{n+1}=\frac{p_{n-1}-q_{n-1}\alpha}{q_{n}\alpha-p_{n}}$ (cf. Corollary 16). Hence, $\alpha_{n+1}+\beta_{n+1}=\frac{p_{n-1}q_{n}-p_{n}q_{n-1}}{q_{n}(q_{n}\alpha-p_{n})}=\frac{(-1)^{n}}{q_{n}(q_{n}\alpha-p_{n})}$ (by Corollary 15). This proves the proposition. ∎

Therefore, the proposition says that $\ell(\alpha)=\limsup\limits_{n\to\infty}(\alpha_{n}+\beta_{n})$. From the dynamical point of view, we consider the symbolic space $\Sigma=(\mathbb{N}^{*})^{\mathbb{Z}}=:\Sigma^{-}\times\Sigma^{+}=(\mathbb{N}^{*})^{\mathbb{Z}^{-}}\times(\mathbb{N}^{*})^{\mathbb{N}}$ equipped with the left shift dynamics $\sigma:\Sigma\to\Sigma$, $\sigma((a_{n})_{n\in\mathbb{Z}}):=(a_{n+1})_{n\in\mathbb{Z}}$ and the height function $f:\Sigma\to\mathbb{R}$, $f((a_{n})_{n\in\mathbb{Z}})=[a_{0};a_{1},a_{2},\dots]+[0;a_{-1},a_{-2},\dots]$. Then, the proposition above implies that

[TABLE]

where $\alpha=[a_{0};a_{1},a_{2},\dots]$ and $\underline{\theta}=(\dots,a_{-1},a_{0},a_{1},\dots)$. In particular,

[TABLE]

where $\ell(\underline{\theta}):=\limsup\limits_{n\to+\infty}f(\sigma^{n}(\underline{\theta}))$.

Also, the Markov spectrum has a similar description:

[TABLE]

where $m(\underline{\theta}):=\sup\limits_{n\in\mathbb{Z}}f(\sigma^{n}(\underline{\theta}))$.

Remark 22*.*

A geometrical interpretation of $\sigma:\Sigma\to\Sigma$ is provided by the so-called Gauss map999From Number Theory rather than Differential Geometry.:

[TABLE]

for $0<x\leq 1$.

Indeed, $G([0;a_{1},a_{2},\dots])=[0;a_{2},\dots]$, so that $\sigma:\Sigma\to\Sigma$ is a symbolic version of the natural extension of $G$.

Furthermore, the identification $(\dots,a_{-1},a_{0},a_{1},\dots)\simeq([0;a_{-1},a_{-2},\dots],[a_{0};a_{1},a_{2},\dots])=(y,x)$ allows us to write the height function as $f((a_{n})_{n\in\mathbb{Z}})=x+y$.

Perron’s dynamical interpretation of the Lagrange and Markov spectra is the starting point of many results about $L$ and $M$ which are not so easy to guess from their definitions:

Exercise 23**.**

Show that $L\subset M$ are closed subsets of $\mathbb{R}$.

Remark 24*.*

$M-L\neq\emptyset$: for example, Freiman [6] proved in 1968 that

[TABLE]

has the property that $3.118120178\simeq m(s)\in M-L$. (Here $\overline{\theta_{1}\dots\theta_{n}}$ means infinite repetition of the block $\theta_{1}\dots\theta_{n}$.)

Also, Freiman [7] showed in 1973 that $m(s_{n})\in M-L$ and $m(s_{n})\to m(s_{\infty})\simeq 3.293044265\in M-L$ where

[TABLE]

for $n\geq 4$, and

[TABLE]

1.5. Digression: Lagrange spectrum and cusp excursions on the modular surface

The Lagrange spectrum is related to the values of a certain height function $H$ along the orbits of the geodesic flow $g_{t}$ on the (unit cotangent bundle to) the modular surface: indeed, we will show that

[TABLE]

Remark 25*.*

This fact is not surprising to experts: the Gauss map appears naturally by quotienting out the weak-stable manifolds of $g_{t}$ as observed by Artin, Series, Arnoux, … (see, e.g., [1]).

An unimodular lattice in $\mathbb{R}^{2}$ has the form $g(\mathbb{Z}^{2})$, $g\in SL(2,\mathbb{Z})$, and the stabilizer in $SL(2,\mathbb{R})$ of the standard lattice $\mathbb{Z}^{2}$ is $SL(2,\mathbb{Z})$. In particular, the space of unimodular lattices in $\mathbb{R}^{2}$ is $SL(2,\mathbb{R})/SL(2,\mathbb{Z})$.

As it turns out, $SL(2,\mathbb{R})/SL(2,\mathbb{Z})$ is the unit cotangent bundle to the modular surface $\mathbb{H}/SL(2,\mathbb{Z})$ (where $\mathbb{H}=\{z\in\mathbb{C}:\textrm{Im}(z)>0\}$ is the hyperbolic upper-half plane and \left(\begin{array}[]{cc}a&b\\ c&d\end{array}\right)\in SL(2,\mathbb{R}) acts on $z\in\mathbb{H}$ via \left(\begin{array}[]{cc}a&b\\ c&d\end{array}\right)\cdot z=\frac{az+b}{cz+d}).

The geodesic flow of the modular surface is the action of g_{t}=\left(\begin{array}[]{cc}e^{t}&0\\ 0&e^{-t}\end{array}\right) on $SL(2,\mathbb{R})/SL(2,\mathbb{Z})$. The stable and unstable manifolds of $g_{t}$ are the orbits of the stable and unstable horocycle flows h_{s}=\left(\begin{array}[]{cc}1&0\\ s&1\end{array}\right) and u_{s}=\left(\begin{array}[]{cc}1&s\\ 0&1\end{array}\right): indeed, this follows from the facts that $g_{t}h_{s}=h_{se^{-2t}}g_{t}$ and $g_{t}u_{s}=u_{se^{t}}g_{t}$.

The set of holonomy (or primitive) vectors of $\mathbb{Z}^{2}$ is

[TABLE]

In general, the set $\textrm{Hol}(X)$ of holonomy vectors of $X=g(\mathbb{Z}^{2})$, $g\in SL(2,\mathbb{Z})$, is

[TABLE]

The systole $\textrm{sys}(X)$ of $X=g(\mathbb{Z}^{2})$ is

[TABLE]

Remark 26*.*

By Mahler’s compactness criterion [19], $X\mapsto\frac{1}{\textrm{sys}(X)}$ is a proper function on $SL(2,\mathbb{R})/SL(2,\mathbb{Z})$.

Remark 27*.*

For later reference, we write $\textrm{Area}(v):=|\textrm{Re}(v)|\cdot|\textrm{Im}(v)|$ for the area of the rectangle in $\mathbb{R}^{2}$ with diagonal $v=(\textrm{Re}(v),\textrm{Im}(v))\in\mathbb{R}^{2}$.

Proposition 28**.**

The forward geodesic flow orbit of $X\in SL(2,\mathbb{R})/SL(2,\mathbb{Z})$ does not go straight to infinity (i.e., $\textrm{sys}(g_{t}(X))\to 0$ as $t\to+\infty$) if and only if there is no vertical vector in $\textrm{Hol}(X)$. In this case, there are (unique) parameters $s,t,\alpha\in\mathbb{R}$ such that

[TABLE]

Proof.

By unimodularity, any $X=g(\mathbb{Z}^{2})$ has a single short holonomy vector. Since $g_{t}$ contracts vertical vectors and expands horizontal vectors for $t>0$, we have that $\textrm{sys}(g_{t}(X))\to 0$ as $t\to+\infty$ if and only if $\textrm{Hol}(X)$ contains a vertical vector.

By Iwasawa decomposition, there are (unique) parameters $s,t,\theta\in\mathbb{R}$ such that $X=h_{s}g_{t}r_{\theta}$, where r_{\theta}=\left(\begin{array}[]{cc}\cos\theta&-\sin\theta\\ \sin\theta&\cos\theta\end{array}\right). Since $\cos\theta\neq 0$ when $\textrm{Hol}(X)$ contains no vertical vector and, in this situation,

[TABLE]

we see that $X=h_{s+e^{-2t}\tan\theta}\cdot g_{t+\log\cos\theta}\cdot u_{-\tan\theta}(\mathbb{Z}^{2})$ (because $h_{s}g_{t}r_{\theta}=h_{s}g_{t}h_{\tan\theta}g_{\log\cos\theta}u_{-\tan\theta}=h_{s+e^{-2t}\tan\theta}\cdot g_{t+\log\cos\theta}\cdot u_{-\tan\theta}$). This ends the proof of the proposition. ∎

Proposition 29**.**

Let $X=h_{s}g_{t}u_{-\alpha}(\mathbb{Z}^{2})$ be an unimodular lattice without vertical holonomy vectors. Then,

[TABLE]

Remark 30*.*

This proposition says that the dynamical quantity $\limsup\limits_{T\to+\infty}\frac{2}{\textrm{sys}(g_{T}(X))^{2}}$ does not depend on the “weak-stable part” $h_{s}g_{t}$ (but only on $\alpha$) and it can be computed without dynamics by simply studying almost vertical holonomy vectors in $X$.

Proof.

Note that $\textrm{Area}(g_{t}(v))=\textrm{Area}(v)$ for all $t\in\mathbb{R}$ and $v\in\mathbb{R}^{2}$. Since $\textrm{Area}(v)=\frac{\|g_{t(v)}(v)\|^{2}}{2}$ for $t(v):=\frac{1}{2}\log\frac{|\textrm{Im}(v)|}{|\textrm{Re}(v)|}$, the equality $\limsup\limits_{\begin{subarray}{c}|\textrm{Im}(v)|\to\infty\\ v\in\textrm{Hol}(X)\end{subarray}}\frac{1}{\textrm{Area}(v)}=\limsup\limits_{T\to+\infty}\frac{2}{\textrm{sys}(g_{T}(X))^{2}}$ follows.

The relation $g_{T}h_{s}=h_{se^{-2T}}g_{T}$ and the continuity of the systole function imply that $\limsup\limits_{T\to+\infty}\frac{2}{\textrm{sys}(g_{T}(X))^{2}}$ depends only on $\alpha$. Because any $v\in\textrm{Hol}(u_{-\alpha}(\mathbb{Z}^{2}))$ has the form $v=(p-q\alpha,q)=u_{-\alpha}(p,q)$ with $(p,q)\in\textrm{Hol}(\mathbb{Z}^{2})$, the equality $\limsup\limits_{\begin{subarray}{c}|\textrm{Im}(v)|\to\infty\\ v\in\textrm{Hol}(X)\end{subarray}}\frac{1}{\textrm{Area}(v)}=\ell(\alpha)$. ∎

In summary, the previous proposition says that the Lagrange spectrum $L$ coincides with

[TABLE]

where $H(y)=\frac{2}{\textrm{sys}(y)^{2}}$ is a (proper) height function and $g_{t}$ is the geodesic flow on $SL(2,\mathbb{R})/SL(2,\mathbb{Z})$.

Remark 31*.*

Several number-theoretical problems translate into dynamical questions on the modular surface: for example, Zagier [30] showed that the Riemann hypothesis is equivalent to a certain speed of equidistribution of $u_{s}$-orbits on $SL(2,\mathbb{R})/SL(2,\mathbb{Z})$.

1.6. Hall’s ray and Freiman’s constant

In 1947, M. Hall [9] proved that:

Theorem 32** (Hall).**

The half-line $[6,+\infty)$ is contained in $L$.

This result motivates the following nomenclature: the biggest half-line $[c_{F},+\infty)\subset L(\subset M)$ is called Hall’s ray.

In 1975, G. Freiman [8] determined Hall’s ray:

Theorem 33** (Freiman).**

$c_{F}=4+\frac{253589820+283798\sqrt{462}}{491993569}\simeq 4.527829566...$**

The constant $c_{F}$ is called Freiman’s constant.

Let us sketch the proof of Hall’s theorem based on the following lemma:

Lemma 34** (Hall).**

Denote by $C(4):=\{[0;a_{1},a_{2},\dots]\in\mathbb{R}:a_{i}\in\{1,2,3,4\}\,\,\forall\,i\in\mathbb{N}\}$. Then,

[TABLE]

Remark 35*.*

The reader can find a proof of this lemma in Cusick-Flahive’s book [3]. Interestingly enough, some of the techniques in the proof of Hall’s lemma were rediscovered much later (in 1979) in the context of Dynamical Systems by Newhouse [26] (in the proof of his gap lemma).

Remark 36*.*

$C(4)$ is a dynamical Cantor set101010See Subsections 2.2 and 2.3 below. whose Hausdorff dimension is $>1/2$ (see Remark 48 below). In particular, $C(4)\times C(4)$ is a planar Cantor set of Hausdorff dimension $>1$ and Hall’s lemma says that its image $f(C(4)\times C(4))=C(4)+C(4)$ under the the projection $f(x,y)=x+y$ contains an interval. Hence, Hall’s lemma can be thought as a sort of “particular case” of Marstrand’s theorem [18] (ensuring that typical projections of planar sets with Hausdorff dimension $>1$ has positive Lebesgue measure).

For our purposes, the specific form $C(4)+C(4)$ is not important: the key point is that $C(4)+C(4)$ is an interval of length $>1$.

Indeed, given $6\leq\ell<\infty$, Hall’s lemma guarantees the existence of $c_{0}\in\mathbb{N}$, $5\leq c_{0}\leq\ell$ such that $\ell-c_{0}\in C(4)+C(4)$. Thus,

[TABLE]

with $a_{i},b_{i}\in\{1,2,3,4\}$ for all $i\in\mathbb{N}$.

Define

[TABLE]

Since $c_{0}\geq 5>4\geq a_{i},b_{i}$ for all $i\in\mathbb{N}$, Perron’s characterization of $\ell(\alpha)$ implies that

[TABLE]

This proves Theorem 32.

1.7. Statement of Moreira’s theorem

Our discussion so far can be summarized as follows:

• •

$L\cap(-\infty,3)=M\cap(-\infty,3)=\{k_{1}<k_{2}<\dots<k_{n}<\dots\}$ is an explicit discrete set;

• •

$L\cap[c_{F},\infty)=M\cap[c_{F},\infty)$ is an explicit ray.

Moreira’s theorem [21] says that the intermediate parts $L\cap[3,c_{F}]$ and $M\cap[3,c_{F}]$ of the Lagrange and Markov spectra have an intricate structure:

Theorem 37** (Moreira).**

For each $t\in\mathbb{R}$, the sets $L\cap(-\infty,t)$ and $M\cap(-\infty,t)$ have the same Hausdorff dimension, say $d(t)\in[0,1]$.

Moreover, the function $t\mapsto d(t)$ is continuous, $d(3+\varepsilon)>0$ for all $\varepsilon>0$ and $d(\sqrt{12})=1$ (even though $\sqrt{12}=3.4641...<4.5278...=c_{F}$).

Remark 38*.*

Many results about $L$ and $M$ are dynamical111111I.e., they involve Perron’s characterization of $L$ and $M$, the study of Gauss map and/or the geodesic flow on the modular surface, etc.. In particular, it is not surprising that many facts about $L$ and $M$ have counterparts for dynamical Lagrange and Markov spectra121212I.e., the collections of “records” of height functions along orbits of dynamical systems.: for example, Hall ray or intervals in dynamical Lagrange spectra were found by Parkkonen-Paulin [27], Hubert-Marchese-Ulcigrai [11] and Moreira-Romaña [23], and the continuity result in Moreira’s theorem 37 was recently extended by Cerqueira, Moreira and the author in [2].

Before entering into the proof of Moreira’s theorem, let us close this section by briefly recalling the notion of Hausdorff dimension.

1.8. Hausdorff dimension

The $s$-Hausdorff measure $m_{s}(X)$ of a subset $X\subset\mathbb{R}^{n}$ is

[TABLE]

The Hausdorff dimension of $X$ is

[TABLE]

Remark 39*.*

There are many notions of dimension in the literature: for example, the box-counting dimension of $X$ is $\lim\limits_{\delta\to 0}\frac{\log N_{X}(\delta)}{\log(1/\delta)}$ where $N_{X}(\delta)$ is the smallest number of boxes of side lengths $\leq\delta$ needed to cover $X$. As an exercise, the reader is invited to show that the Hausdorff dimension is always smaller than or equal to the box-counting dimension.

The following exercise (whose solution can be found in Falconer’s book [5]) describes several elementary properties of the Hausdorff dimension:

Exercise 40**.**

Show that:

• (a)

if $X\subset Y$, then $HD(X)\leq HD(Y)$;

• (b)

$HD(\bigcup\limits_{i\in\mathbb{N}}X_{i})=\sup\limits_{i\in\mathbb{N}}HD(X_{i})$; in particular, $HD(X)=0$ whenever $X$ is a countable set (such as $X=\{p\}$ or $X=\mathbb{Q}^{n}$);

• (c)

if $f:X\to Y$ is $\alpha$-Hölder continuous131313I.e., for some constant $C>0$, one has $|f(x)-f(x^{\prime})|\leq C|x-x^{\prime}|^{\alpha}$ for all $x,x^{\prime}\in X$., then $\alpha\cdot HD(f(X))\leq HD(X)$;

• (d)

$HD(\mathbb{R}^{n})=n$* and, more generally, $HD(X)=m$ when $X\subset\mathbb{R}^{n}$ is a smooth $m$-dimensional submanifold.*

Example 41**.**

Cantor’s middle-third set $C=\{\sum\limits_{i=1}^{\infty}\frac{a_{i}}{3^{i}}:a_{i}\in\{0,2\}\,\,\forall\,i\in\mathbb{N}\}$ has Hausdorff dimension $\frac{\log 2}{\log 3}\in(0,1)$: see Falconer’s book [5] for more details.

Using item (c) of Exercise 40 above, we have the following corollary of Moreira’s theorem 37:

Corollary 42** (Moreira).**

The function $t\mapsto HD(L\cap(-\infty,t))$ is not $\alpha$-Hölder continuous for any $\alpha>0$.

Proof.

By Theorem 37, $d$ maps $L\cap[3,3+\varepsilon]$ to the non-trivial interval $[0,d(3+\varepsilon)]$ for any $\varepsilon>0$. By item (c) of Exercise 40, if $t\mapsto d(t)=HD(L\cap(-\infty,t))$ were $\alpha$-Hölder continuous for some $\alpha>0$, then it would follow that

[TABLE]

for all $\varepsilon>0$. On the other hand, Theorem 37 (and item (b) of Exercise 40) also says that

[TABLE]

In summary, $0<\alpha\leq\lim\limits_{\varepsilon\to 0}d(3+\varepsilon)=0$, a contradiction. ∎

2. Proof of Moreira’s theorem

2.1. Strategy of proof of Moreira’s theorem

Roughly speaking, the continuity of $d(t)=HD(L\cap(-\infty,t))$ is proved in four steps:

• •

if $0<d(t)<1$, then for all $\eta>0$ there exists $\delta>0$ such that $L\cap(-\infty,t-\delta)$ can be “approximated from inside” by $K+K^{\prime}=f(K\times K^{\prime})$ where $K$ and $K^{\prime}$ are Gauss-Cantor sets with $HD(K)+HD(K^{\prime})=HD(K\times K^{\prime})>(1-\eta)d(t)$ (and $f(x,y)=x+y$);

• •

by Moreira’s dimension formula (derived from profound works of Moreira and Yoccoz on the geometry of Cantor sets), we have that

[TABLE]

• •

thus, if $0<d(t)<1$, then for all $\eta>0$ there exists $\delta>0$ such that

[TABLE]

hence, $d(t)$ is lower semicontinuous;

• •

finally, an elementary compactness argument shows the upper semicontinuity of $d(t)$.

Remark 43*.*

This strategy is purely dynamical because the particular forms of the height function $f$ and the Gauss map $G$ are not used. Instead, we just need the transversality of the gradient of $f$ to the stable and unstable manifolds (vertical and horizontal axis) and the non-essential affinity of Gauss-Cantor sets. (See [2] for more explanations.)

In the remainder of this section, we will implement (a version of) this strategy in order to deduce the continuity result in Theorem 37.

2.2. Dynamical Cantor sets

A dynamically defined Cantor set $K\subset\mathbb{R}$ is

[TABLE]

where $I_{1},\dots,I_{k}$ are pairwise disjoint compact intervals, and $\psi:I_{1}\cup\dots\cup I_{k}\to I$ is a $C^{r}$-map from $I_{1}\cup\dots\cup I_{k}$ to its convex hull $I$ such that:

• •

$\psi$ is uniformly expanding: $|\psi^{\prime}(x)|>1$ for all $x\in I_{1}\cup\dots\cup I_{k}$;

• •

$\psi$ is a (full) Markov map: $\psi(I_{j})=I$ for all $1\leq j\leq k$.

Remark 44*.*

Dynamical Cantor sets are usually defined with a weaker Markov condition, but we stick to this definition for simplicity.

Example 45**.**

Cantor’s middle-third set $C=\{\sum\limits_{i=1}^{\infty}\frac{a_{i}}{3^{i}}:a_{i}\in\{0,2\}\,\,\forall\,i\in\mathbb{N}\}$ is

[TABLE]

where $\psi:[0,1/3]\cup[2/3,1]\to[0,1]$ is given by

[TABLE]

Remark 46*.*

A dynamical Cantor set is called affine when $\psi|_{I_{j}}$ is affine for all $j$. In this language, Cantor’s middle-third set is an affine dynamical Cantor set.

Example 47**.**

Given $A\geq 2$, let $C(A):=\{[0;a_{1},a_{2},\dots]:1\leq a_{i}\leq A\,\,\forall\,i\in\mathbb{N}\}$. This is a dynamical Cantor set associated to Gauss map: for example,

[TABLE]

where $I_{1}$ and $I_{2}$ are the intervals depicted below.

Remark 48*.*

Hensley [10] showed that

[TABLE]

and Jenkinson-Pollicott [13], [14] used thermodynamical formalism methods to obtain that

[TABLE]

[TABLE]

2.3. Gauss-Cantor sets

The set $C(A)$ above is a particular case of Gauss-Cantor set:

Definition 49**.**

Given $B=\{\beta_{1},\dots,\beta_{l}\}$, $l\geq 2$, a finite, primitive141414I.e., $\beta_{i}$ doesn’t begin by $\beta_{j}$ for all $i\neq j$. alphabet of finite words $\beta_{j}\in(\mathbb{N}^{*})^{r_{j}}$, the Gauss-Cantor set $K(B)\subset[0,1]$ associated to $B$ is

[TABLE]

Example 50**.**

$C(A)=K(\{1,\dots,A\})$.

Exercise 51**.**

Show that any Gauss-Cantor set $K(B)$ is dynamically defined.151515Hint: For each word $\beta_{j}\in(\mathbb{N}^{*})^{r_{j}}$, let $I(\beta_{j})=\{[0;\beta_{j},a_{1},\dots]:a_{i}\in\mathbb{N}\,\,\forall\,i\}=I_{j}$ and $\psi|_{I_{j}}:=G^{r_{j}}$ where $G(x)=\{1/x\}$ is the Gauss map.

From the symbolic point of view, $B=\{\beta_{1},\dots,\beta_{l}\}$ as above induces a subshift

[TABLE]

Also, the corresponding Gauss-Cantor is $K(B)=\{[0;\gamma]:\gamma\in\Sigma^{+}(B)\}$ where $\Sigma^{+}(B)=\pi^{+}(\Sigma(B))$ and $\pi^{+}:\Sigma\to\Sigma^{+}$ is the natural projection (related to local unstable manifolds of the left shift map on $\Sigma$).

For later use, denote by $B^{T}=\{\beta^{T}:\beta\in B\}$ the transpose of $B$, where $\beta^{T}:=(a_{n},\dots,a_{1})$ for $\beta=(a_{1},\dots,a_{n})$.

The following proposition (due to Euler) is proved in Appendix B:

Proposition 52** (Euler).**

If $[0;\beta]=\frac{p_{n}}{q_{n}}$, then $[0;\beta^{T}]=\frac{r_{n}}{q_{n}}$.

A striking consequence of this proposition is:

Corollary 53**.**

$HD(K(B))=HD(K(B^{T}))$.

Sketch of proof.

The lengths of the intervals $I(\beta)=\{[0;\beta,a_{1},\dots]:a_{i}\in\mathbb{N}\,\,\forall\,i\}$ in the construction of $K(B)$ depend only on the denominators of the partial quotients of $[0;\beta]$. Therefore, we have from Proposition 52 that $K(B)$ and $K(B^{T})$ are Cantor sets constructed from intervals with same lengths, and, a fortiori, they have the Hausdorff dimension. ∎

Remark 54*.*

This corollary is closely related to the existence of area-preserving natural extensions of Gauss map (see [1]) and the coincidence of stable and unstable dimensions of a horseshoe of an area-preserving surface diffeomorphism (see [20]).

2.4. Non-essentially affine Cantor sets

We say that

[TABLE]

is non-essentially affine if there is no global conjugation $h\circ\psi\circ h^{-1}$ such that all branches

[TABLE]

are affine maps of the real line.

Equivalently, if $p\in K$ is a periodic point of $\psi$ of period $k$ and $h:I\to I$ is a diffeomorphism of the convex hull $I$ of $I_{1}\cup\dots\cup I_{r}$ such that $h\circ\psi^{k}\circ h^{-1}$ is affine161616Such a diffeomorphism $h$ linearizing one branch of $\psi$ always exists by Poincaré’s linearization theorem. on $h(J)$ where $J$ is the connected component of the domain of $\psi^{k}$ containing $p$, then $K$ is non-essentially affine if and only if $(h\circ\psi\circ h^{-1})^{\prime\prime}(x)\neq 0$ for some $x\in h(K)$.

Proposition 55**.**

Gauss-Cantor sets are non-essentially affine.

Proof.

The basic idea is to explore the fact that the second derivative of a non-affine Möbius transformation never vanishes.

More concretely, let $B=\{\beta_{1},\dots,\beta_{m}\}$, $\beta_{j}\in(\mathbb{N}^{*})^{r_{j}}$, $1\leq j\leq m$. For each $\beta_{j}$, let

[TABLE]

be the fixed point of the branch $\psi|_{I_{j}}=G^{r_{j}}$ of the expanding map $\psi$ naturally171717Cf. Exercise 51. defining the Gauss-Cantor set $K(B)$.

By Corollary 16, $\psi|_{I_{j}}(x)=\frac{q^{(j)}_{r_{j}-1}x-p^{(j)}_{r_{j}-1}}{p^{(j)}_{r_{j}}-q^{(j)}_{r_{j}}x}$ where $\frac{p^{(j)}_{k}}{q^{(j)}_{k}}=[0;b^{(j)}_{1},\dots,b^{(j)}_{k}]$ and $\beta_{j}=(b^{(j)}_{1},\dots,b^{(j)}_{r_{j}})$.

Note that the fixed point $x_{j}$ of $\psi|_{I_{j}}$ is the positive solution of the second degree equation

[TABLE]

In particular, $x_{j}$ is a quadratic surd.

For each $1\leq j\leq k$, the Möbius transformation $\psi|_{I_{j}}$ has a hyperbolic fixed point $x_{j}$. It follows (from Poincaré linearization theorem) that there exists a Möbius transformation

[TABLE]

linearizing $\psi|_{I_{j}}$, i.e., $\alpha_{j}(x_{j})=x_{j}$, $\alpha^{\prime}(x_{j})=1$ and $\alpha_{j}\circ(\psi|_{I_{j}})\circ\alpha_{j}^{-1}$ is an affine map.

Since non-affine Möbius transformations have non-vanishing second derivative, the proof of the proposition will be complete once we show that $\alpha_{1}\circ(\psi|_{I_{2}})\circ\alpha_{1}^{-1}$ is not affine. So, let us suppose by contradiction that $\alpha_{1}\circ(\psi|_{I_{2}})\circ\alpha_{1}^{-1}$ is affine. In this case, $\infty$ is a common fixed point of the (affine) maps $\alpha_{1}\circ(\psi|_{I_{2}})\circ\alpha_{1}^{-1}$ and $\alpha_{1}\circ(\psi|_{I_{1}})\circ\alpha_{1}^{-1}$, and, a fortiori, $\alpha_{1}^{-1}(\infty)=-d_{1}/c_{1}$ is a common fixed point of $\psi|_{I_{1}}$ and $\psi|_{I_{2}}$. Thus, the second degree equations

[TABLE]

would have a common root. This implies that these polynomials coincide (because they are polynomials in $\mathbb{Z}[x]$ which are irreducible181818Thanks to the fact that their roots $x_{1},x_{2}\notin\mathbb{Q}$.) and, hence, their other roots $x_{1}$, $x_{2}$ must coincide, a contradiction. ∎

2.5. Moreira’s dimension formula

The Hausdorff dimension of projections of products of non-essentially affine Cantor sets is given by the following formula:

Theorem 56** (Moreira).**

Let $K$ and $K^{\prime}$ be two $C^{2}$ dynamical Cantor sets. If $K$ is non-essentially affine, then the projection $f(K\times K^{\prime})=K+K^{\prime}$ of $K\times K^{\prime}$ under $f(x,y)=x+y$ has Hausdorff dimension

[TABLE]

Remark 57*.*

This statement is a particular case of Moreira’s dimension formula (which is sufficient for our current purposes because Gauss-Cantor sets are non-essentially affine).

The proof of this result is out of the scope of these notes: indeed, it depends on the techniques introduced in two works (from 2001 and 2010) by Moreira and Yoccoz [24], [25] such as fine analysis of limit geometries and renormalization operators, “recurrence on scales”, “compact recurrent sets of relative configurations”, and Marstrand’s theorem. We refer the reader to [22] for more details.

Remark 58*.*

Moreira’s dimension formula is coherent with Hall’s Lemma 34: in fact, since $HD(C(4))>1/2$, it is natural that $HD(C(4)+C(4))=1$.

2.6. First step towards Moreira’s theorem 37: projections of Gauss-Cantor sets

Let $\Sigma(B)\subset(\mathbb{N}^{*})^{\mathbb{Z}}$ be a complete shift of finite type. Denote by $\ell(\Sigma(B))$, resp. $m(\Sigma(B))$, the pieces of the Lagrange, resp. Markov, spectrum generated by $\Sigma(B)$, i.e.,

[TABLE]

where $\ell(\underline{\theta})=\limsup\limits_{n\to\infty}f(\sigma^{n}(\underline{\theta}))$, $m(\underline{\theta})=\sup\limits_{n\in\mathbb{Z}}f(\sigma^{n}(\underline{\theta}))$, $f((\theta_{i})_{i\in\mathbb{Z}})=[\theta_{0};\theta_{1},\dots]+[0;\theta_{-1},\dots]$ and $\sigma((\theta_{i})_{i\in\mathbb{Z}})=(\theta_{i+1})_{i\in\mathbb{Z}}$ is the shift map.

The following proposition relates the Hausdorff dimensions of the pieces of the Langrange and Markov spectra associated to $\Sigma(B)$ and the projection $f(K(B)\times K(B^{T}))$:

Proposition 59**.**

One has $HD(\ell(\Sigma(B)))=HD(m(\Sigma(B)))=\min\{1,2\cdot HD(K(B))\}$.

Sketch of proof.

By definition,

[TABLE]

where $R\in\mathbb{N}$ is the largest entry among all words of $B$.

Thus, $HD(\ell(\Sigma(B)))\leq HD(m(\Sigma(B)))\leq HD(K(B))+HD(K(B^{T}))$. By Corollary 53, it follows that

[TABLE]

By Moreira’s dimension formula (cf. Theorem 56), our task is now reduced to show that for all $\varepsilon>0$, there are “replicas” $K$ and $K^{\prime}$ of Gauss-Cantor sets such that

[TABLE]

In this direction, let us order $B$ and $B^{T}$ by declaring that $\gamma<\gamma^{\prime}$ if and only if $[0;\gamma]<[0;\gamma^{\prime}]$.

Given $\varepsilon>0$, we can replace if necessary $B$ and/or $B^{T}$ by $B^{n}=\{\gamma_{1}\dots\gamma_{n}:\gamma_{i}\in B\,\,\forall\,i\}$ and/or $(B^{T})^{n}$ for some large $n=n(\varepsilon)\in\mathbb{N}$ in such a way that

[TABLE]

where $A^{*}:=\{\min A,\max A\}$. Indeed, this holds because the Hausdorff dimension of a Gauss-Cantor set $K(A)$ associated to an alphabet $A$ with a large number of words does not decrease too much after removing only two words from $A$.

We expect the values of $\ell$ on $((B^{T})^{*})^{\mathbb{Z}^{-}}\times(B^{*})^{\mathbb{N}}$ to decrease because we removed the minimal and maximal elements of $B$ and $B^{T}$ (and, in general, $[a_{0};a_{1},a_{2},\dots]<[b_{0};b_{1},b_{2},\dots]$ if and only if $(-1)^{k}(a_{k}-b_{k})<0$ where $k$ is the smallest integer with $a_{k}\neq b_{k}$).

In particular, this gives some control on the values of $\ell$ on $((B^{T})^{*})^{\mathbb{Z}^{-}}\times(B^{*})^{\mathbb{N}}$, but this does not mean that $K(B^{*})+K((B^{T})^{*})\subset\ell(\Sigma(B))$.

We overcome this problem by studying replicas of $K(B^{*})$ and $K((B^{T})^{*})$. More precisely, let $\widetilde{\theta}=(\dots,\widetilde{\gamma}_{0},\widetilde{\gamma}_{1},\dots)\in\Sigma(B)$, $\widetilde{\gamma}_{i}\in B$ for all $i\in\mathbb{Z}$, such that

[TABLE]

is attained at a position in the block $\widetilde{\gamma}_{0}$.

By compactness, there exists $\eta>0$ and $m\in\mathbb{N}$ such that any

[TABLE]

with $\gamma_{i}\in B^{*}$ for all $i>m$ and $\gamma_{i}\in(B^{T})^{*}$ for all $i<-m$ satisfies:

• •

$m(\theta)$ is attained in a position in the central block $(\widetilde{\gamma}_{-m},\dots,\widetilde{\gamma}_{0},\dots,\widetilde{\gamma}_{m})$;

• •

$f(\sigma^{n}(\theta))<m(\theta)-\eta$ for any non-central position $n$.

By exploring these properties, it is possible to enlarge the central block to get a word called $\tau^{\#}=(a_{-N_{1}},\dots,a_{0},\dots,a_{N_{2}})$ in Moreira’s paper [21] such that the replicas

[TABLE]

and

[TABLE]

of $K(B^{*})$ and $K((B^{T})^{*})$ have the desired properties that

[TABLE]

and

[TABLE]

This completes our sketch of proof of the proposition. ∎

2.7. Second step towards Moreira’s theorem 37: upper semi-continuity

Let $\Sigma_{t}:=\{\theta\in(\mathbb{N}^{*})^{\mathbb{Z}}:m(\theta)\leq t\}$ for $3\leq t<5$.

Our long term goal is to compare $\Sigma_{t}$ with its projection $K_{t}^{+}:=\{[0;\gamma]:\gamma\in\pi^{+}(\Sigma_{t})\}$ on the unstable part (where $\pi^{+}:(\mathbb{N}^{*})^{\mathbb{Z}}\to(\mathbb{N}^{*})^{\mathbb{N}}$ is the natural projection).

Given $\alpha=(a_{1},\dots,a_{n})$, its unstable scale $r^{+}(\alpha)$ is

[TABLE]

where $I^{+}(\alpha)$ is the interval with extremities $[0;a_{1},\dots,a_{n}]$ and $[0;a_{1},\dots,a_{n}+1]$.

Denote by

[TABLE]

and

[TABLE]

Remark 60*.*

By symmetry (i.e., replacing $\gamma$’s by $\gamma^{T}$’s), we can define $K^{-}_{t}$, $r^{-}(\alpha)$, etc.

For later use, we observe that the unstable scales have the following behaviour under concatenations of words:

Exercise 61**.**

Show that $r^{+}(\alpha\beta k)\geq r^{+}(\alpha)+r^{+}(\beta)$ for all $\alpha$, $\beta$ finite words and for all $k\in\{1,2,3,4\}$.

In particular, since the family of intervals

[TABLE]

covers $K_{t}^{+}$, it follows from Exercise 61 that

[TABLE]

for all $r,s\in\mathbb{N}$ and, hence, the sequence $(4\#C^{+}(t,r))_{r\in\mathbb{N}}$ is submultiplicative.

So, the box-counting dimension (cf. Remark 39) $\Delta^{+}(t)$ of $K_{t}^{+}$ is

[TABLE]

An elementary compactness argument shows that the upper-semicontinuity of $\Delta^{+}(t)$:

Proposition 62**.**

The function $t\mapsto\Delta^{+}(t)$ is upper-semicontinuous.

Proof.

For the sake of contradiction, assume that there exist $\eta>0$ and $t_{0}$ such that $\Delta^{+}(t)>\Delta^{+}(t_{0})+\eta$ for all $t>t_{0}$.

By definition, this means that there exists $r_{0}\in\mathbb{N}$ such that

[TABLE]

for all $r\geq r_{0}$ and $t>t_{0}$.

On the other hand, $C^{+}(t,r)\subset C^{+}(s,r)$ for all $t\leq s$ and, by compactness, $C^{+}(t_{0},r)=\bigcap\limits_{t>t_{0}}C^{+}(t,r)$. Thus, if $r\to\infty$ and $t\to t_{0}$, the inequality of the previous paragraph would imply that

[TABLE]

a contradiction. ∎

2.8. Third step towards Moreira’s theorem 37: lower semi-continuity

The main result of this subsection is the following theorem allowing us to “approximate from inside” $\Sigma_{t}$ by Gauss-Cantor sets.

Theorem 63**.**

Given $\eta>0$ and $3\leq t<5$ with $d(t):=HD(L\cap(-\infty,t))>0$, we can find $\delta>0$ and a Gauss-Cantor set $K(B)$ associated to $\Sigma(B)\subset\{1,2,3,4\}^{\mathbb{Z}}$ such that

[TABLE]

This theorem allows us to derive the continuity statement in Moreira’s theorem 37:

Corollary 64**.**

$\Delta^{-}(t)=\Delta^{+}(t)$* is a continuous function of $t$ and $d(t)=\min\{1,2\cdot\Delta^{+}(t)\}$.*

Proof.

By Corollary 53 and Theorem 63, we have that

[TABLE]

Also, a “symmetric” estimate holds after exchanging the roles of $\Delta^{-}$ and $\Delta^{+}$. Hence, $\Delta^{-}(t)=\Delta^{+}(t)$. Moreover, the inequality above says that $\Delta^{-}(t)=\Delta^{+}(t)$ is a lower-semicontinuous function of $t$. Since we already know that $\Delta^{+}(t)$ is an upper-semicontinuous function of $t$ thanks to Proposition 62, we conclude that $t\mapsto\Delta^{-}(t)=\Delta^{+}(t)$ is continuous. Finally, by Proposition 59, from $\Sigma(B)\subset\Sigma_{t-\delta}$, we also have that

[TABLE]

Since $d(t)\leq\min\{1,\Delta^{+}(t)+\Delta^{-}(t)\}$ (because $\Sigma_{t}\subset\pi^{-}(\Sigma_{t})\times\pi^{+}(\Sigma_{t})$), the proof is complete. ∎

Let us now sketch the construction of the Gauss-Cantor sets $K(B)$ approaching $\Sigma_{t}$ from inside.

Sketch of proof of Theorem 63.

Fix $r_{0}\in\mathbb{N}$ large enough so that

[TABLE]

for all $r\geq r_{0}$.

Set $B_{0}:=C^{+}(t,r_{0})$, $k=8(\#B_{0})^{2}\lceil 80/\eta\rceil$ and

[TABLE]

It is not hard to show that $\widetilde{B}$ has a significant cardinality in the sense that

[TABLE]

In particular, one can use this information to prove that $HD(K(\widetilde{B}))$ is not far from $\Delta^{+}(t)$, i.e.

[TABLE]

Unfortunately, since we have no control on the values of $m$ on $\Sigma(\widetilde{B})$, there is no guarantee that $\Sigma(\widetilde{B})\subset\Sigma_{t-\delta}$ for some $\delta>0$.

We can overcome this issue with the aid of the notion of left-good and right-good positions. More concretely, we say that $1\leq j\leq k$ is a right-good position of $\beta=(\beta_{1},\dots,\beta_{k})\in\widetilde{B}$ whenever there are two elements $\beta^{(s)}=\beta_{1}\dots\beta_{j}\beta_{j+1}^{(s)}\dots\beta_{k}^{(s)}\in\widetilde{B}$, $s\in\{1,2\}$ such that

[TABLE]

Similarly, $1\leq j\leq k$ is a left-good position $\beta=(\beta_{1},\dots,\beta_{k})\in\widetilde{B}$ whenever there are two elements $\beta^{(s)}=\beta_{1}\dots\beta_{j}\beta_{j+1}^{(s)}\dots\beta_{k}^{(s)}\in\widetilde{B}$, $s\in\{3,4\}$ such that

[TABLE]

Furthermore, we say that $1\leq j\leq k$ is a good position of $\beta=(\beta_{1},\dots,\beta_{k})\in\widetilde{B}$ when it is both a left-good and a right-good position.

Since there are at most two choices of $\beta_{j}\in B_{0}$ when $\beta_{1},\dots,\beta_{j-1}$ are fixed and $j$ is a right-good position, one has that the subset

[TABLE]

of excellent words in $\widetilde{B}$ has cardinality

[TABLE]

We expect the values of $m$ on $\Sigma(\mathcal{E})$ to decrease because excellent words have many good positions. Also, the Hausdorff dimension of $K(\mathcal{E})$ is not far from $\Delta^{+}(t)$ thanks to the estimate above on the cardinality of $\mathcal{E}$. However, there is no reason for $\Sigma(\mathcal{E})\subset\Sigma_{t-\delta}$ for some $\delta>0$ because an arbitrary concatenation of words in $\mathcal{E}$ might not belong to $\Sigma_{t}$.

At this point, the idea is to build a complete shift $\Sigma(B)\subset\Sigma_{t-\delta}$ from $\mathcal{E}$ with the following combinatorial argument. Since $\beta=(\beta_{1},\dots,\beta_{k})\in\mathcal{E}$ has $9k/10$ good positions, we can find good positions $1\leq i_{1}\leq i_{2}\leq\dots\leq i_{\lceil 2k/5\rceil}\leq k-1$ such that $i_{s}+2\leq i_{s+1}$ for all $1\leq s\leq\lceil 2k/5\rceil-1$ and $i_{s}+1$ are also good positions for all $1\leq s\leq\lceil 2k/5\rceil$. Because $k:=8(\#B_{0})^{2}\lceil 80/\eta\rceil$, the pigeonhole principle reveals that we can choose positions $j_{1}\leq\dots\leq j_{3(\#B_{0})^{2}}$ and words $\widehat{\beta}_{j_{1}},\widehat{\beta}_{j_{1}+1},\dots,\widehat{\beta}_{j_{3(\#B_{0})^{2}}},\widehat{\beta}_{j_{3(\#B_{0})^{2}}+1}\in B_{0}$ such that $j_{s}+2\lceil 80/\eta\rceil\leq j_{s+1}$ for all $s<3(\#B_{0})^{2}$ and the subset

[TABLE]

of excellent words with prescribed subwords $\widehat{\beta}_{j_{s}}$, $\widehat{\beta}_{j_{s}+1}$ at the good positions $j_{s}$, $j_{s}+1$ has cardinality

[TABLE]

Next, we convert $X$ into the alphabet $B$ of an appropriate complete shift with the help of the projections $\pi_{a,b}:X\to B_{0}^{j_{b}-j_{a}}$, $\pi_{a,b}(\beta_{1},\dots,\beta_{k})=(\beta_{j_{a}+1},\beta_{j_{a}+2},\dots,\beta_{j_{b}})$. More precisely, an elementary counting argument shows that we can take $1\leq a<b\leq 3(\#B_{0})^{2}$ such that $\widehat{\beta}_{j_{a}}=\widehat{\beta}_{j_{b}}$, $\widehat{\beta}_{j_{a}+1}=\widehat{\beta}_{j_{b}+1}$, and the image $\pi_{a,b}(X)$ of some projection $\pi_{a,b}$ has a significant cardinality

[TABLE]

From these properties, we get an alphabet $B=\pi_{a,b}(X)$ whose words concatenate in an appropriate way (because $\widehat{\beta}_{j_{a}}=\widehat{\beta}_{j_{b}}$, $\widehat{\beta}_{j_{a}+1}=\widehat{\beta}_{j_{b}+1}$), the Hausdorff dimension of $K(B)$ is $HD(K(B))>(1-\eta)\Delta^{+}(t)$ (because $\#B>(\#B_{0})^{(1-\tfrac{\eta}{4})(j_{b}-j_{a})}$ and $j_{b}-j_{a}>2\lceil\tfrac{80}{\eta}\rceil$), and $\Sigma(B)\subset\Sigma_{t-\delta}$ for some $\delta>0$ (because the features of good positions forces the values of $m$ on $\Sigma(B)$ to decrease). This completes our sketch of proof. ∎

2.9. End of proof of Moreira’s theorem 37

By Corollary 64, the function

[TABLE]

is continuous. Moreover, an inspection of the proof of Corollary 64 shows that we have also proved the equality $HD(M\cap(-\infty,t))=HD(L\cap(-\infty,t))$.

Therefore, our task is reduced to prove that $d(3+\varepsilon)>0$ for all $\varepsilon>0$ and $d(\sqrt{12})=1$.

The fact that $d(3+\varepsilon)>0$ for any $\varepsilon$ uses explicit sequences $\theta_{m}\in\{1,2\}^{\mathbb{Z}}$ such that $\lim\limits_{m\to\infty}m(\theta_{m})=3$ in order to exhibit non-trivial Cantor sets in $M\cap(-\infty,3+\varepsilon)$. More precisely, consider191919This choice of $\theta_{m}$ is motivated by the discussion in Chapter 1 of Cusick-Flahive book [3]. the periodic sequences

[TABLE]

where $\overline{a_{1}\dots a_{k}}:=\dots a_{1}\dots a_{k}\,\,a_{1}\dots a_{k}\dots$. Since the sequence $\theta_{\infty}=\overline{1},2,2,\overline{1}$ has the property that $m(\theta_{\infty})=[2;\overline{1}]+[0;2,\overline{1}]=3$, and $|[a_{0};a_{1},\dots,a_{n},b_{1},\dots]-[a_{0};a_{1},\dots,a_{n},c_{1},\dots]|<\frac{1}{2^{n-1}}$ in general202020See Lemma 2 in Chapter 1 of [3]., we have that the alphabet $B_{m}$ consisting of the two words $2\underbrace{1\dots 1}_{2m\textrm{ times}}2$ and $2\underbrace{1\dots 1}_{2m+2\textrm{ times}}2$ satisfies

[TABLE]

Thus, $d(3+\tfrac{1}{2^{m}})=HD(M\cap(-\infty,3+\frac{1}{2^{m}}))\geq HD(\Sigma(B_{m}))=2\cdot HD(K(B_{m}))>0$ for all $m\in\mathbb{N}$.

Finally, the fact that $d(\sqrt{12})=1$ follows from Corollary 64 and Remark 48. Indeed, Perron showed that $m(\theta)\leq\sqrt{12}$ if and only if $\theta\in\{1,2\}^{\mathbb{Z}}$ (see the proof of Lemma 7 in Chapter 1 of Cusick-Flahive book [3]). Thus, $K_{\sqrt{12}}^{+}=C(2)$. By Corollary 64, it follows that

[TABLE]

Since Remark 48 tells us that $HD(C(2))>1/2$, we conclude that $d(\sqrt{12})=1$.

Appendix A Proof of Hurwitz theorem

Given $\alpha\notin\mathbb{Q}$, we want to show that the inequality

[TABLE]

has infinitely many rational solutions.

In this direction, let $\alpha=[a_{0};a_{1},\dots]$ be the continued fraction expansion of $\alpha$ and denote by $[a_{0};a_{1},\dots,a_{n}]=p_{n}/q_{n}$. We affirm that, for every $\alpha\notin\mathbb{Q}$ and every $n\geq 1$, we have

[TABLE]

for some $\frac{p}{q}\in\{\frac{p_{n-1}}{q_{n-1}},\frac{p_{n}}{q_{n}},\frac{p_{n+1}}{q_{n+1}}\}$.

Remark 65*.*

Of course, this last statement provides infinitely many solutions to the inequality $\left|\alpha-\frac{p}{q}\right|\leq\frac{1}{\sqrt{5}q^{2}}$. So, our task is reduced to prove the affirmation above.

The proof of the claim starts by recalling Perron’s Proposition 21:

[TABLE]

where $\alpha_{n+1}:=[a_{n+1};a_{n+2},\dots]$ and $\beta_{n+1}=\frac{q_{n-1}}{q_{n}}=[0;a_{n},\dots,a_{1}]$.

For the sake of contradiction, suppose that the claim is false, i.e., there exists $k\geq 1$ such that

[TABLE]

Since $\sqrt{5}<3$ and $a_{m}\leq\alpha_{m}+\beta_{m}$ for all $m\geq 1$, it follows from (A.1) that

[TABLE]

If $a_{m}=2$ for some $k\leq m\leq k+2$, then (A.2) would imply that $\alpha_{m}+\beta_{m}\geq 2+[0;2,1]=2+\frac{1}{3}>\sqrt{5}$, a contradiction with our assumption (A.1).

So, our hypothesis (A.1) forces

[TABLE]

Denoting by $x=\frac{1}{\alpha_{k+2}}$ and $y=\beta_{k+1}=q_{k-1}/q_{k}\in\mathbb{Q}$, we have from (A.3) that

[TABLE]

By plugging this into (A.1), we obtain

[TABLE]

On one hand, (A.4) implies that

[TABLE]

Thus,

[TABLE]

and, a fortiori, $y(\sqrt{5}-y)\geq 1$, i.e.,

[TABLE]

On the other hand, (A.4) implies that

[TABLE]

Hence,

[TABLE]

and, a fortiori, $(1+y)(\sqrt{5}-1-y)\geq 1$, i.e.,

[TABLE]

It follows from (A.5) and (A.6) that $y=(\sqrt{5}-1)/2$, a contradiction because $y=\beta_{k+1}=q_{k-1}/q_{k}\in\mathbb{Q}$. This completes the argument.

Appendix B Proof of Euler’s remark

Denote by $[0;a_{1},a_{2},\dots,a_{n}]=\frac{p(a_{1},\dots,a_{n})}{q(a_{1},\dots,a_{n})}=\frac{p_{n}}{q_{n}}$. It is not hard to see that

[TABLE]

From this formula, we see that $q(a_{1},\dots,a_{n})$ is a sum of the following products of elements of $\{a_{1},\dots,a_{n}\}$. First, we take the product $a_{1}\dots a_{n}$ of all $a_{i}$’s. Secondly, we take all products obtained by removing any pair $a_{i}a_{i+1}$ of adjacent elements. Then, we iterate this procedure until no pairs can be omitted (with the convention that if $n$ is even, then the empty product gives $1$). This rule to describe $q(a_{1},\dots,a_{n})$ was discovered by Euler.

It follows immediately from Euler’s rule that $q(a_{1},\dots,a_{n})=q(a_{n},\dots,a_{1})$. This proves Proposition 52.