Markov constant

In number theory, specifically in Diophantine approximation theory, the Markov constant M(\alpha) of an irrational number \alpha is the factor for which Dirichlet's approximation theorem can be improved for \alpha.

History and motivation

Certain numbers can be approximated well by certain rationals; specifically, the convergents of the continued fraction are the best approximations by rational numbers having denominators less than a certain bound. For example, the approximation \pi\approx\frac{22}{7} is the best rational approximation among rational numbers with denominator up to 56.^[1] Also, some numbers can be approximated more readily than others. Dirichlet proved in 1840 that the least readily approximable numbers are the rational numbers, in the sense that for every irrational number there exists infinitely many rational numbers approximating it to a certain degree of accuracy that only finitely many such rational approximations exist for rational numbers. Specifically, he proved that for any number \alpha there are infinitely many pairs of relatively prime numbers (p,q) such that \left|\alpha - \frac{p}{q}\right| < \frac{1}{q^2} if and only if \alpha is irrational.

51 years later, Hurwitz further improved Dirichlet's approximation theorem by a factor of √5,^[2] improving the right-hand side from 1/q^2 to 1/\sqrt{5}q^2 for irrational numbers:

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

The above result is best possible since the golden ratio \phi is irrational but if we replace √5 by any larger number in the above expression then we will only be able to find finitely many rational numbers that satisfy the inequality for \alpha=\phi.

Furthermore, he showed that among the irrational numbers, the least readily approximable numbers are those of the form \frac{a\phi+b}{c\phi+d} where \phi is the golden ratio, a,b,c,d\in\Z and ad-bc=\pm1.^[3] (These numbers are said to be equivalent to \phi.) If we omit these numbers, just as we omitted the rational numbers in Dirichlet's theorem, then we can increase the number √5 to 2√2. Again this new bound is best possible in the new setting, but this time the number √2, and numbers equivalent to it, limits the bound.^[3] If we don't allow those numbers then we can again increase the number on the right hand side of the inequality from 2√2 to √221/5,^[3] for which the numbers equivalent to \frac{1+\sqrt{221}}{10} limit the bound. The numbers generated show how well these numbers can be approximated; this can be seen as a property of the real numbers.

However, instead of considering Hurwitz's theorem (and the extensions mentioned above) as a property of the real numbers except certain special numbers, we can consider it as a property of each excluded number. Thus, the theorem can be interpreted as "numbers equivalent to \phi, √2 or \frac{1+\sqrt{221}}{10} are among the least readily approximable irrational numbers." This leads us to consider how accurately each number can be approximated by rationals - specifically, by how much can the factor in Dirichlet's approximation theorem be increased to from 1 for that specific number.

Definition

Mathematically, the Markov constant of irrational \alpha is defined as M(\alpha)=\sup \left\{\lambda\in\R : \left\vert \alpha - \frac{p}{q} \right\vert<\frac{1}{\lambda q^2} \text{ has infinitely many solutions for }p,q\in\N \right\}.^[4] If the set does not have an upper bound we define M(\alpha)=\infty.

Alternatively, it can be defined as \limsup_{k\to\infty}\frac{1}{k^2\left\vert \alpha-\frac{f(k)}{k} \right\vert} where f(k) is defined as the closest integer to \alpha k.

Properties and results

Hurwitz's theorem implies that M(\alpha)\ge\sqrt{5} for all \alpha\in\R-\Q.

If \alpha = [a_0; a_1, a_2, ...] is its continued fraction expansion then M(\alpha)=\limsup_{k\to\infty}{([a_{k+1}; a_{k+2}, a_{k+3}, ...] + [0; a_{k}, a_{k-1}, ...,a_2,a_1])}.^[4]

From the above, if p=\limsup_{k\to\infty}{a_k} then p<M(\alpha)<p+2. This implies that M(\alpha)=\infty if and only if (a_k) is not bounded. In particular, M(\alpha)<\infty if \alpha is a quadratic irrational number. In fact, the lower bound for M(\alpha) can be strengthened to M(\alpha)\ge\sqrt{p^2+4}, the tightest possible.^[5]

The values of \alpha for which M(\alpha)<3 are families of quadratic irrationalities having the same period (but at different offsets), and the values of M(\alpha) for these \alpha are limited to Lagrange numbers. There are uncountably many numbers for which M(\alpha)=3, no two of which have the same ending; for instance, for each number \alpha = [\underbrace{1;1,...,1}_{r_1},2,2,\underbrace{1,1,...,1}_{r_2},2,2,\underbrace{1,1,...,1}_{r_3},2,2,...] where r_1<r_2<r_3<\cdots, M(\alpha)=3.^[4]

If \beta=\frac{p\alpha+q}{r\alpha+s} where p,q,r,s\in\Z then M(\beta)\ge\frac{M(\alpha)}{\left\vert ps-rq \right\vert}.^[6] In particular if \left\vert ps-rq \right\vert=1 then M(\beta)=M(\alpha).^[7]

The set L=\{M(\alpha)\mid\alpha\in\R-\Q\} forms the Lagrange spectrum. It contains the interval [F,\infty] where F is Freiman's constant.^[7] Hence, if m>F\approx4.52783 then there exists irrational \alpha whose Markov constant is m.

Numbers having a Markov constant less than 3

Burger et al. (2002)^[8] provides a formula for which the quadratic irrationality \alpha_n whose Markov constant is the n^th Lagrange number:

\alpha_n=\frac{2u-3m_n+\sqrt{9m_n^2-4}}{2m_n} where m_n is the n^th Markov number, and u is the smallest positive integer such that m_n\mid u^2+1.

Nicholls (1978)^[9] provides a geometric proof of this (based on circles tangent to each other), providing a method that these numbers can be systematically found.

Examples

Markov constant of two numbers

Since \frac{\sqrt{10}}{2}=[1;\overline{1,1,2}],

\begin{align} M\left ( \frac{\sqrt{10}}{2} \right ) & = \max([1;\overline{2,1,1}]+[0;\overline{1,2,1}],[1;\overline{1,2,1}]+[0;\overline{2,1,1}],[2;\overline{1,1,2}]+[0;\overline{1,1,2}]) \\ & = \max\left ( \frac{2\sqrt{10}}{3},\frac{2\sqrt{10}}{3},\sqrt{10} \right ) \\ & = \sqrt{10}. \end{align}

As e = [2; 1, 2, 1, 1, 4, 1, 1, 6, 1, \ldots, 1, 2n, 1, \ldots], M(e)=\infty because the continued fraction representation of e is unbounded.

Numbers α_n having Markov constant less than 3

Consider n=6; Then m_n=34. By trial and error it can be found that u=13. Then

\begin{align} \alpha_6 & = \frac{2u-3m_6+\sqrt{9m_6^2-4}}{2m_6} \\[6pt] & = \frac{-76+\sqrt{10400}}{68} \\[6pt] &= \frac{-19+5\sqrt{26}}{17} \\[6pt] &=[0;\overline{2,1,1,1,1,1,1,2}]. \end{align}

See also

• Irrationality measure
• Lagrange number
• Continued fraction
• Lagrange spectrum

References

1. ^ Fernando, Suren L. (27 July 2001). "A063673 (Denominators of sequence {3/1, 13/4, 16/5, 19/6, 22/7, 179/57, 201/64, 223/71, 245/78, 267/85, 289/92, 311/99, 333/106, ... } of approximations to Pi with increasing denominators, where each approximation is an improvement on its predecessors.)". The On-Line Encyclopedia of Integer Sequences. Retrieved 2 December 2019.
2. ^ Hurwitz, A. (1891). "Ueber die angenäherte Darstellung der Irrationalzahlen durch rationale Brüche (On the approximate representation of irrational numbers by rational fractions)" (in German). Mathematische Annalen. 39 (2): 279–284. doi:10.1007/BF01206656. S2CID 119535189 contains the actual proof in German.
3. ^ Weisstein, Eric W. (25 November 2019). "Hurwitz's Irrational Number Theorem". Wolfram Mathworld. Retrieved 2 December 2019.
4. ^ LeVeque, William (1977). Fundamentals of Number Theory. Addison-Wesley Publishing Company, Inc. pp. 251–254. ISBN 0-201-04287-8.
5. ^ Hancl, Jaroslav (January 2016). "Second basic theorem of Hurwitz". Lithuanian Mathematical Journal. 56: 72–76. doi:10.1007/s10986-016-9305-4. S2CID 124639896
6. ^ Pelantová, Edita; Starosta, Štěpán; Znojil, Miloslav (2016). "Markov constant and quantum instabilities". Journal of Physics A: Mathematical and Theoretical. 49 (15). arXiv:1510.02407. Bibcode:2016JPhA...49o5201P. doi:10.1088/1751-8113/49/15/155201. S2CID 119161523
7. ^ Hazewinkel, Michiel (1990). Encyclopaedia of Mathematics. Springer Science & Business Media. p. 106. ISBN 978-1-55608-005-0.
8. ^ Burger, Edward B.; Folsom, Amanda; Pekker, Alexander; Roengpitya, Rungporn; Snyder, Julia (2002). "On a quantitative refinement of the Lagrange spectrum". Acta Arithmetica. 102 (1): 59–60. Bibcode:2002AcAri.102...55B. doi:10.4064/aa102-1-5
9. ^ Nicholls, Peter (1978). "Diophantine Approximation via the Modular Group". Journal of the London Mathematical Society. 17: 11–17. Second Series. doi:10.1112/jlms/s2-17.1.11