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Diophantine Approximation, Lagrange and Markov Spectra, and Dynamical Cantor Sets

Carlos Matheus
Carlos Gustavo Moreira

Communicated by Notices Associate Editor William McCallum

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1. Diophantine Approximation

The seminal works of Diophantus of Alexandria (circa AD 250) on rational approximations to the solutions of certain algebraic equations began the important subfield of number theory called Diophantine approximation. Among the basic problems in this topic, one has the question of finding rational numbers approximating a given real number in such a way that the denominator is not “big” and the error is “small.”

1.1. Rational approximations of

The first few decimal digits of the number are well known: . By definition, this provides some rational approximations of like and . Nonetheless, these fractions are certainly not answers to the Diophantine problem posed above because we can get better approximations with smaller denominators: for instance, Archimedes (circa 250 BC) knew that

and it is possible to check that

1.2. Dirichlet’s pigeonhole principle

The example of the number makes us wonder how small can be when the denominator varies in a fixed range . A preliminary answer comes from the following elementary remark. Recall that any real number lies between two consecutive integers, namely , where is the integer part of . Therefore, given and , we can find such that , i.e.,

In 1841, Dirichlet used his famous pigeonhole principle to significantly improve upon the elementary statement in the previous paragraph: more concretely, for any irrational number , one has that

Indeed, given , Dirichlet considered how the numbers , , …, , are distributed across the elements of the partition into equal intervals. By the pigeonhole principle, two fractional parts, say , , must lie in the same interval, say , so that and, a fortiori, there exists such that

Figure 1.

1.3. Hurwitz theorem

In 1891, Hurwitz proved⁠Footnote1 Actually, this result was first established by Korkine and Zolotarev in 1873. that Dirichlet’s theorem is essentially optimal as far as all irrational numbers are concerned: one has

for all irrational numbers , and

for all .

2. Classical Spectra

Despite the almost optimality of Dirichlet’s theorem, we can ask whether it can be improved for individual irrational numbers by inquiring about the nature of the best constant among the quantities such that

i.e.,

The Lagrange spectrum is the collection of finiteFootnote2 It is possible to show that for Lebesgue almost every . Hence, the Lagrange spectrum tries to encode Diophantine properties of irrational numbers beyond the probabilistic dominant regime. best constants of Diophantine approximation, i.e.,

In this setting, the Hurwitz theorem says that the minimum of is . The Lagrange spectrum is an amazingly complex object. In this section we recount the history of results about it, including those indicated in Figure 1. We refer to the slightly expanded version of this survey article in arXiv:2105.01449 [math.NT] for a more detailed discussion on some of these results.

2.1. Beginning of the classical spectra

The Lagrange spectrum was systematically studied in connection with the theory of binary quadratic forms by Markov in 1879. In fact, the quantity is the value of the binary quadratic form at the integral point , so that the Lagrange spectrum is somewhat related to the Markov spectrum of finite best constants

of Diophantine approximations of real, indefinite, binary quadratic forms with positive discriminant . In this context, Markov proved that

where is a Markov number, i.e., the largest coordinate of a solution of the Markov–Hurwitz equation

2.1.1. Fermat’s descent on Markov’s cubic. The Markov–Hurwitz equation determines a cubic surface whose integral points are called Markov triples. Since the Markov–Hurwitz equation is quadratic on a given variable (when we freeze the other two variables), the cubic surface has a rich group of automorphisms made available by swapping roots of those quadratic equations: besides permuting the coordinates, we can replace by , , or without leaving . The last three automorphisms are called Vieta involutions and they were used by Markov to produce a descent argument showing that any Markov triple can be obtained from the fundamental solution after applying a sequence of permutations of coordinates and Vieta involutions.

In fact, it is not hard to see that a Markov triple with falls into two categories: either or . In the first case, it turns out that or . In the second case, applying the Vieta involution with yields a Markov triple , where is now the largest number. (Details are left to the reader.) By permuting the coordinates and repeating this argument finitely many times, we see that a sequence of Vieta involutions and permutations of coordinates allows us to convert the Markov triple into , as desired.

2.1.2. The Markov tree. The descent argument above permits us to organize all ordered Markov triples , , into the so-called Markov tree whose branches connect ordered Markov triples deduced from each other by a Vieta involution (up to permutation of coordinates).

The knowledge of Markov’s tree permits us to write down the first few elements of : since the first few Markov triples are , , , , and , we have that the first few Markov numbers⁠Footnote3 The attentive reader certainly noticed that some of these numbers are part of Fibonacci’s sequence and this is not a coincidence: it is possible to check that is a Markov triple for all . are , , , , and , so that

2.1.3. Beyond the Markov tree…. The Markov tree and numbers are fascinating objects. For instance, it was conjectured by Frobenius in 1913 that Markov triples , , are actually determined by the Markov number (cf. Bombieri’s survey article Bom07).

Also, Zagier Zag82 showed that the number of Markov numbers below is

where is an explicit constant, and, more recently, Baragar Bar94 and Gamburd–Magee–Ronan GMR19 studied the general problem of counting integral points on the Markov–Hurwitz varieties of the form

where , , and are integers.

Moreover, the Markov triples are related to lengths of simple closed geodesics on a certain hyperbolic once-punctured torus: in fact, the commutator subgroup of is an index subgroup generated by and ; the quotient is the unit cotangent bundle of a hyperbolic once-punctured torus whose simple closed geodesics correspond to the elements in a pair of generators of ; the hyperbolic lengths of these geodesics have the form , so that they are related to Markov triples because Fricke proved that any generating pair of satisfies

i.e., is a Markov triple.

Furthermore, it is known (cf. Gol03) that the level sets of the function parametrize the elements of the -character variety⁠Footnote4 Naively speaking, the -character variety of a topological surface of genus with punctures is the set of equivalence classes of representations modulo the natural action of by conjugation. of once-punctured torii and each Markov triple produces an integral point of .

Finally, Bourgain–Gamburd–Sarnak BGS16 investigated the family of graphs (indexed by the set of prime numbers ) obtained by applying Vieta involutions and permutation of coordinates to the solutions in to the Markov–Hurwitz equation . In this setting, they showed has a giant component in the sense that for all , and they used the technology involved in the proof of this statement to establish that almost all Markov numbers are composite, i.e.,

as . Also, they conjectured that the graphs are connected⁠Footnote5 The connectedness of for all large was very recently established by Chen; cf. the arxiv preprint arXiv:2011.12940. and they form an expander family.⁠Footnote6 That is, there is a uniform spectral gap for the adjacency matrices of these graphs.

2.2. Continued fractions

The definition of the Lagrange spectrum suggests that we can study provided there is a method to find the best rational approximations of a given irrational number (such as and for ).

As it turns out, one can guess the best rational approximations for out of its continued fraction expansion. More precisely, given an irrational number , let , so that . We define recursively and for all . In this context, we say that has continued fraction expansion

and we denote by

the convergents of . For example, , so that

It is possible to prove that provides the best rational approximations to in the sense that every convergent is within of , every convergent is closer to than any rational number with smaller denominator, and every rational approximation that is within of is a convergent.

In particular, the best constant

of Diophantine approximation for depends only on its convergents, i.e.,

2.3. Perron’s definition of the spectra

The basic formula

where , led Perron to propose in 1921 the following dynamical interpretation of . Let be the (noncompact) symbolic space of bi-infinite sequences of nonzero natural numbers. The left shift map is the dynamical system given by

In this language, the Lagrange spectrum is the set of finite asymptotic records of heights of the orbits of with respect to the (proper) height function , , i.e.,

To see this, embed a continued fraction in by filling in to the left of the 0 position with any sequence of nonzero natural numbers. Asymptotically these numbers do not contribute to the height as they are shifted to the left.

Interestingly enough, one can use the classical reduction theory of binary quadratic forms (due to Lagrange and Gauss) to prove that the Markov spectrum is the set of finite absolute records of heights of the orbits of with respect to , i.e.,

From these dynamical characterizations of and , Perron deduced that

if and only if ;

;

.

Moreover, one can use this dynamical point of view to prove that

and

Here eventually periodic means eventually periodic on both sides (perhaps with different periods). Thus, are closed subsets of the real line.

2.4. Dynamics on the modular surface

The shift map can be thought of as an invertible map extending the Gauss map , . Indeed, the definitions imply that the Gauss map acts on continued fraction expansions by left-shift on half-infinite sequences of natural numbers:

Using the well-known link (due to Artin, Cohn, Series, Arnoux, …) between the Gauss map and the geodesic flow on the unit cotangent bundle to the modular surface (cf. Arn94), one can also describe the Lagrange spectrum as the set of finite asymptotic records of the heights of the orbits of a continuous-time, smooth dynamical system, namely,

where is a certain (proper) function.⁠Footnote7 By thinking of as the space of unimodular lattices in , one has , where is the systole of , .

2.5. The end of the classical spectra

The expression of the height function in Perron’s definition of the spectra suggests that and are related to arithmetic sums of Cantor sets of real numbers whose continued fraction expansions have restricted digits.

In other terms, the study of projections of products of certain Cantor sets under the function , , should provide some insights into the fine structures of and .

This idea was explored by Hall in 1947 to show that contains the half-line . For this sake, Hall considered the continued fraction Cantor set and he established that

is an interval of length . This fact implies that given , one can find such that and , say

with for all . Thus, the irrational number with continued fraction expansion