Jahresber Dtsch Math-VerDOI 10.1365/s13291-014-0090-x

B O O K R E V I E W

Martin Aigner: “Markov’s Theorem and 100 Yearsof the Uniqueness Conjecture”Springer-Verlag, 2013, 257 pp.

Yann Bugeaud

© Deutsche Mathematiker-Vereinigung and Springer-Verlag Berlin Heidelberg 2014

The present book tells the story of a celebrated theorem andan intriguing conjecture: Markov’s theorem from 1879 andthe uniqueness conjecture formulated by Frobenius in 1913.The author takes the opportunity to look at this theorem andthis conjecture from many different viewpoints, includingFarey fractions, the modular and the free groups, the hyper-bolic plane and combinatorics on words. He offers a journeythrough the mathematical world around Markov’s theorem ina leisurely and relaxed style, making his book very pleasantto read.

As a starting point, let us recall that every irrational realnumber α can be uniquely expressed has a continued fraction

α = [a0;a1, a2, . . .] = a0 + 1

a1 + 1

a2 + 1

. . .

,

where a0 is the integer part of α and a1, a2, . . . are positive integers, called the partialquotients of α. For n ≥ 1, denoting by pn/qn the rational number [a0;a1, a2, . . . , an],the theory of continued fractions gives that |α − pn/qn| < 1/q2

n . This shows thatthe inequality |α − p/q| < 1/q2 has infinitely many solutions in integers p,q withq ≥ 1. Said differently, every irrational real number can be approximated at order 2 byrational numbers, a result which also follows from Dirichlet’s theorem. An irrationalreal number α is called badly approximable if there exists a positive real number C

Y. Bugeaud (B)Strasbourg, Francee-mail: bugeaud@math.unistra.fr

Y. Bugeaud

such that |α − p/q| > C/q2 holds for all integers p,q with q ≥ 1. Equivalently,α is badly approximable if the sequence of its partial quotients is bounded. The setof badly approximable numbers has zero Lebesgue measure and has full Hausdorffdimension.

The Lagrange number L(α) of α is the supremum over all the real numbers L suchthat |α − p/q| < 1/(Lq2) has infinitely many solutions in integers p,q with q ≥ 1.It satisfies 1 ≤ L(α) < +∞ when α is badly approximable. The Lagrange spectrumis the set

L := {L(α) : α is badly approximable

}.

In view of the equality

L(α) = lim supn→+∞

([an+1;an+2, . . .] + [0;an, . . . , a1]),

which follows from the theory of continued fractions, the smallest element of Lis

√5, the Lagrange number of the Golden ratio (1 + √

5)/2. This was proved byHurwitz, who also established that

√8 is the second smallest element of L. Many

additional results can be found in the monograph [4] by Thomas W. Cusick and MaryE. Flahive.

We are now in position to state Markov’s theorem, that implies that 3 is the small-est limit point of L.

Let M = {1,2,5,13,29,34, . . .} be the sequence of Markov numbers. The La-grange spectrum below 3 is given by

L<3 ={√

9m2 − 4

m: m ∈M

}.

More precisely, there is a sequence of quadratic numbers

γm = am + √9m2 − 4

bm

, m ∈M,

with am,bm integers, whose Lagrange numbers are

L(γm) =√

9m2 − 4

m.

Conversely, every element of L less than 3 is of this form.But what are these Markov numbers? The triples (m1,m2,m3) of positive integers

which are solution of the Markov equation

x21 + x2

2 + x23 = 3x1x2x3

are called Markov triples, and the numbers that appear in such a triple are calledMarkov numbers. Markov’s theorem links Diophantine approximation to Diophan-tine equations in a beautiful and unexpected way.

In an important memoir published in 1913, Frobenius conducted the first in-depthstudy of the Markov numbers and mentioned the uniqueness conjecture, which statesthat every Markov number appears exactly once as the maximum in a Markov triple.An equivalent formulation of the conjecture asserts that if α and β satisfy L(α) =

M. Aigner: “Markov’s Theorem”

L(β) < 3, then the continued fractions of α and β differ by only finitely many partialquotients.

In Chapter 3, the author explains how the Markov triples can be arranged in aninfinite binary tree, the Markov tree, thereby showing that the set M of Markovnumbers is infinite. He describes the Farey table and gives a few easy results on theuniqueness conjecture, including the fact that every Markov number m of the formm = pk or m = 2pk , with k ≥ 1 and p an odd prime, is unique.

In the 1950s, Harvey Cohn noticed that a well-known identity involving tracesof 2 × 2 integral matrices looks very much like Markov’s equation. The Cohn treeis presented in Chapter 4, together with a few more very partial results towards theuniqueness conjecture. The next two chapters are devoted to a more in-depth studyof the modular group SL(2,Z) and the hyperbolic plane.

Combinatorics on words is the subject of Chapters 7 and 8. The author discussesChristoffel words, Sturmian words and mechanical words, and their combinatorialproperties. For further results, an interested reader can consult the book [1] by JeanBerstel, Aaron Lauve, Christophe Reutenauer and Franco V. Saliola.

The proof of Markov’s theorem is given in Chapter 9 and is very much inspiredby Enrico Bombieri’s beautiful expository paper [2]. It combines various results es-tablished in the preceding chapters.

The last chapter presents some more very partial results towards the uniquenessconjecture. A first approach uses techniques from Diophantine approximation and asecond one is based on algebraic number theory. The final theorem, which was provedby Button in 2001, states that every odd Markov number m of the form m = Npk ,with k ≥ 1, N ≤ 1035 and p prime, is unique.

The author has deliberately kept the book on an elementary level. The prerequisitesare limited to basic algebra and some number theory. Anything beyond this level,like the necessary concepts about groups, geometry, combinatorics on words andalgebraic number theory, is well explained in the book. An undergraduate studentwill certainly enjoy this reading and learn a lot. He will see how it can be profitableto look at a problem from many different perspectives.

To conclude, I would like to express the regret that the author mentioned onlyvery briefly (two pages) the theory of quadratic forms, which was, however, one ofthe main motivations of Markov. And let me quote two recent related results. Us-ing deep tools from Diophantine approximation (the Schmidt Subspace Theorem),Pietro Corvaja and Umberto Zannier [3] proved that, given any finite set S of primenumbers, Markov’s equation has only finitely many integer solutions x, y, z such thatall the prime divisors of xy belong to S. Damien Roy [5] established a beautifuland unexpected connection between two apparently distant topics of Diophantine ap-proximation: the Lagrange spectrum and the extremal numbers, which are (countablymany) real numbers enjoying a very specific property regarding their approximationby quadratic numbers.

References

1. Berstel, J., Lauve, A., Reutenauer, C., Saliola, F.V.: Combinatorics on Words. Christoffel Words andRepetitions in Words. CRM Monograph Series, vol. 27. American Mathematical Society, Providence(2009)

Y. Bugeaud

2. Bombieri, E.: Continued fractions and the Markoff tree. Expo. Math. 25(3), 187–213 (2007)3. Corvaja, P., Zannier, U.: On the greatest prime factor of Markov pairs. Rend. Semin. Mat. Univ. Padova

116, 253–260 (2006)4. Cusick, T.W., Flahive, M.E.: The Markoff and Lagrange Spectra. Mathematical Surveys and Mono-

graphs, vol. 30. American Mathematical Society, Providence (1989)5. Roy, D.: Markoff–Lagrange spectrum and extremal numbers. Acta Math. 206, 325–362 (2011)