Markov constant

From HandWiki
Jump to: navigation, search
Markov constant of a number
Template:Markov constant chart
Basic features
Parityeven
DomainIrrational numbers
CodomainLagrange spectrum with [math]\displaystyle{ \infty }[/math]
Period1
 
Specific values
Maxima[math]\displaystyle{ +\infty }[/math]
Minima5
Value at [math]\displaystyle{ \phi }[/math]5
Value at 222
 
 

This function is undefined on rationals; hence, it is not continuous.

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

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 [math]\displaystyle{ \pi\approx\frac{22}{7} }[/math] 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[2] 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[further explanation needed]. Specifically, he proved that for any number [math]\displaystyle{ \alpha }[/math] there are infinitely many pairs of relatively prime numbers [math]\displaystyle{ (p,q) }[/math] such that [math]\displaystyle{ \left|\alpha - \frac{p}{q}\right| \lt \frac{1}{q^2} }[/math] if and only if [math]\displaystyle{ \alpha }[/math] is irrational.

51 years later, Hurwitz further improved Dirichlet's approximation theorem by a factor of 5,[3] improving the right-hand side from [math]\displaystyle{ 1/q^2 }[/math] to [math]\displaystyle{ 1/\sqrt{5}q^2 }[/math] for irrational numbers:

[math]\displaystyle{ \left|\alpha - \frac{p}{q}\right| \lt \frac{1}{\sqrt{5}q^2}. }[/math]

The above result is best possible since the golden ratio [math]\displaystyle{ \phi }[/math] 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 [math]\displaystyle{ \alpha=\phi }[/math].

Furthermore, he showed that among the irrational numbers, the least readily approximable numbers are those of the form [math]\displaystyle{ \frac{a\phi+b}{c\phi+d} }[/math] where [math]\displaystyle{ \phi }[/math] is the golden ratio, [math]\displaystyle{ a,b,c,d\in\Z }[/math] and [math]\displaystyle{ ad-bc=\pm1 }[/math].[4] (These numbers are said to be equivalent to [math]\displaystyle{ \phi }[/math].) If we omit these numbers, just as we omitted the rational numbers in Dirichlet's theorem, then we can improve the number 5 by 22. 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.[4] If we don't allow those numbers then we can again increase the number on the right hand side of the inequality from 22 to 221/5,[4] for which the numbers equivalent to [math]\displaystyle{ \frac{1+\sqrt{221}}{10} }[/math] limits the bound. The numbers generated show how well these numbers can be approximated, can 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 [math]\displaystyle{ \phi }[/math], 2 or [math]\displaystyle{ \frac{1+\sqrt{221}}{10} }[/math] 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 [math]\displaystyle{ \alpha }[/math] is defined as [math]\displaystyle{ M(\alpha)=\sup \{\lambda\in\R|\left\vert \alpha - \frac{p}{q} \right\vert\lt \frac{1}{\lambda q^2} \text{ has infinitely many solutions for }p,q\in\N \} }[/math].[5] If the set does not have an upper bound we define [math]\displaystyle{ M(\alpha)=\infty }[/math].

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

Properties and results

Hurwitz's theorem implies that [math]\displaystyle{ M(\alpha)\ge\sqrt{5} }[/math] for all [math]\displaystyle{ \alpha\in\R }[/math].

If [math]\displaystyle{ \alpha = [a_0; a_1, a_2, ...] }[/math] is its continued fraction expansion then [math]\displaystyle{ M(\alpha)=\limsup_{k\to\infty}{([a_k; a_{k+1}, a_{k+2}, ...] + [0; a_{k-1}, a_{k-2}, ...,a_1,a_0])} }[/math].[5]

From the above, if [math]\displaystyle{ p=\limsup_{k\to\infty}{a_k} }[/math] then [math]\displaystyle{ p\lt M(\alpha)\lt p+2 }[/math]. This implies that [math]\displaystyle{ M(\alpha)=\infty }[/math] if and only if [math]\displaystyle{ (a k) }[/math] is not bounded. In particular, [math]\displaystyle{ M(\alpha)\lt \infty }[/math] if [math]\displaystyle{ \alpha }[/math] is a quadratic irrationality. In fact, the lower bound for [math]\displaystyle{ M(\alpha) }[/math] can be strengthened to [math]\displaystyle{ M(\alpha)\ge\sqrt{p^2+4} }[/math], the tightest possible.[6]

The values of [math]\displaystyle{ \alpha }[/math] for which [math]\displaystyle{ M(\alpha)\lt 3 }[/math] are families of quadratic irrationalities having the same period (but at different offsets), and the values of [math]\displaystyle{ M(\alpha) }[/math] for these [math]\displaystyle{ \alpha }[/math] are limited to Lagrange numbers. There are uncountably many numbers for which [math]\displaystyle{ M(\alpha)=3 }[/math], no two of which have the same ending; for instance, for each number [math]\displaystyle{ \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,...] }[/math] where [math]\displaystyle{ r_1\lt r_2\lt r_3\lt \cdots }[/math], [math]\displaystyle{ M(\alpha)=3 }[/math].[5]

If [math]\displaystyle{ \beta=\frac{p\alpha+q}{r\alpha+s} }[/math] where [math]\displaystyle{ p,q,r,s\in\Z }[/math] then [math]\displaystyle{ M(\beta)\ge\frac{M(\alpha)}{\left\vert ps-rq \right\vert} }[/math].[7] In particular if [math]\displaystyle{ \left\vert ps-rq \right\vert=1 }[/math] them [math]\displaystyle{ M(\beta)=M(\alpha) }[/math].[8]

The set [math]\displaystyle{ L=\{M(\alpha)|\alpha\in\R-\Q\} }[/math] forms the Lagrange spectrum. It contains the interval [math]\displaystyle{ [F,\infty] }[/math] where F is Freiman's constant.[8] Hence, if [math]\displaystyle{ m\gt F\approx4.52783 }[/math] then there exists irrational [math]\displaystyle{ \alpha }[/math] whose Markov constant is [math]\displaystyle{ m }[/math].

Numbers having a Markov constant less than 3

Burger et al. (2002)[9] provides a formula for which the quadratic irrationality [math]\displaystyle{ \alpha_n }[/math] whose Markov constant is the nth Lagrange number:

[math]\displaystyle{ \alpha_n=\frac{2u-3m_n+\sqrt{9m_n^2-4}}{2m_n} }[/math] where [math]\displaystyle{ m_n }[/math] is the nth Markov number, and u is the smallest positive integer such that [math]\displaystyle{ m_n\mid u^2+1 }[/math].

Nicholls (1978)[10] provides a geometric proof of this (based on circles tangent to each other), providing a method that these numbers can be systematically found.

Examples

A demonstration that 10/2 has Markov constant 10, as stated in the example below. This plot graphs y(k) = 1/k2|α-f(αk)/k| against log(k) (the natural log of k) where f(x) is the nearest integer to x. The dots at the top corresponding to an x-axis value of 0.7, 2.5, 4.3 and 6.1 (k=2,12,74,456) are the points for which the limit superior of 10 is approached.

Markov constant of a number

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

[math]\displaystyle{ \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} }[/math]

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

Numbers [math]\displaystyle{ \alpha_n }[/math] having Markov constant less than 3

Consider [math]\displaystyle{ n=6 }[/math]; Then [math]\displaystyle{ m_n=34 }[/math]. By trial and error it can be found that [math]\displaystyle{ u=13 }[/math]. Then

[math]\displaystyle{ \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} }[/math]

See also

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.)". https://oeis.org/A063673. 
  2. Koro (22 March 2013). "Dirichlet's approximation theorem". https://planetmath.org/DirichletsApproximationTheorem. 
  3. 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. https://gdz.sub.uni-goettingen.de/id/PPN235181684_0039.  contains the actual proof in German.
  4. 4.0 4.1 4.2 Weisstein, Eric W. (25 November 2019). "Hurwitz's Irrational Number Theorem". http://mathworld.wolfram.com/HurwitzsIrrationalNumberTheorem.html. 
  5. 5.0 5.1 5.2 LeVeque, William (1977). Fundamentals of Number Theory. Addison-Wesley Publishing Company, Inc.. pp. 251–254. ISBN 0-201-04287-8. 
  6. Hancl, Jaroslav (January 2016). "Second basic theorem of Hurwitz". Lithuanian Mathematical Journal 56: 72–76. doi:10.1007/s10986-016-9305-4. 
  7. Pelantová, Edita; Starosta, Štěpán; Znojil, Miloslav (2016). "Markov constant and quantum instabilities". Journal of Physics A: Mathematical and Theoretical 49 (15): 155201. doi:10.1088/1751-8113/49/15/155201. Bibcode2016JPhA...49o5201P. 
  8. 8.0 8.1 Hazewinkel, Michiel (1990). Encyclopaedia of Mathematics. Springer Science & Business Media. pp. 106. ISBN 9781556080050. 
  9. 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. doi:10.4064/aa102-1-5. Bibcode2002AcAri.102...55B. 
  10. Nicholls, Peter (1978). "Diophantine Approximation via the Modular Group". Journal of the London Mathematical Society. Second Series 17: 11–17.