Markov constant

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 $\backslash pi\backslash approx\backslash frac\{22\}\{7\}$ is the best rational approximation among rational numbers with denominator up to 56.{{Cite web|url=https://oeis.org/A063673|title=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.) |last=Fernando|first=Suren L.|date=27 July 2001|website=The On-Line Encyclopedia of Integer Sequences|access-date=2 December 2019}} 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 $\backslash alpha$ there are infinitely many pairs of relatively prime numbers $(p,q)$ such that if and only if $\backslash alpha$ is irrational.

51 years later, Hurwitz further improved Dirichlet's approximation theorem by a factor of {{radic|5}},{{cite journal|last=Hurwitz|first=A.|author-link=Adolf Hurwitz|year=1891|title=Ueber die angenäherte Darstellung der Irrationalzahlen durch rationale Brüche (On the approximate representation of irrational numbers by rational fractions)|journal=Mathematische Annalen|language=de|url=https://gdz.sub.uni-goettingen.de/id/PPN235181684_0039 |volume=39|issue=2|pages=279–284|doi=10.1007/BF01206656|jfm=23.0222.02|s2cid=119535189 }} contains the actual proof in German. improving the right-hand side from $1/q^2$ to $1/\backslash sqrt\{5\}q^2$ for irrational numbers:

:

The above result is best possible since the golden ratio $\backslash phi$ is irrational but if we replace {{radic|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 $\backslash alpha=\backslash phi$.

Furthermore, he showed that among the irrational numbers, the least readily approximable numbers are those of the form $\backslash frac\{a\backslash phi+b\}\{c\backslash phi+d\}$ where $\backslash phi$ is the golden ratio, $a,b,c,d\backslash in\backslash Z$ and $ad-bc=\backslash pm1$.{{Cite web|url=http://mathworld.wolfram.com/HurwitzsIrrationalNumberTheorem.html|title=Hurwitz's Irrational Number Theorem|last=Weisstein|first=Eric W.|date=25 November 2019|website=Wolfram Mathworld|access-date=2 December 2019}} (These numbers are said to be equivalent to $\backslash phi$.) If we omit these numbers, just as we omitted the rational numbers in Dirichlet's theorem, then we can increase the number {{radic|5}} to 2{{radic|2}}. Again this new bound is best possible in the new setting, but this time the number square root of 2, and numbers equivalent to it, limits the bound. If we don't allow those numbers then we can again increase the number on the right hand side of the inequality from 2{{radic|2}} to {{radic|221}}/5, for which the numbers equivalent to $\backslash frac\{1+\backslash sqrt\{221\}\}\{10\}$ 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 $\backslash phi$, {{radic|2}} or $\backslash frac\{1+\backslash sqrt\{221\}\}\{10\}$ 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.

Mathematically, the Markov constant of irrational $\backslash alpha$ is defined as .{{Cite book|title=Fundamentals of Number Theory|last=LeVeque|first=William|publisher=Addison-Wesley Publishing Company, Inc.|year=1977|isbn=0-201-04287-8|pages=251–254}} If the set does not have an upper bound we define $M(\backslash alpha)=\backslash infty$.

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

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

If $\backslash alpha\; =\; [a\_0;\; a\_1,\; a\_2,\; ...]$ is its continued fraction expansion then $M(\backslash alpha)=\backslash limsup\_\{k\backslash to\backslash infty\}\{([a\_\{k+1\};\; a\_\{k+2\},\; a\_\{k+3\},\; ...]\; +\; [0;\; a\_\{k\},\; a\_\{k-1\},\; ...,a\_2,a\_1])\}$.

From the above, if $p=\backslash limsup\_\{k\backslash to\backslash infty\}\{a\_k\}$ then

The values of $\backslash alpha$ for which are families of quadratic irrationalities having the same period (but at different offsets), and the values of $M(\backslash alpha)$ for these $\backslash alpha$ are limited to Lagrange numbers. There are uncountably many numbers for which $M(\backslash alpha)=3$, no two of which have the same ending; for instance, for each number $\backslash alpha\; =\; [\backslash underbrace\{1;1,...,1\}\_\{r\_1\},2,2,\backslash underbrace\{1,1,...,1\}\_\{r\_2\},2,2,\backslash underbrace\{1,1,...,1\}\_\{r\_3\},2,2,...]$ where

If $\backslash beta=\backslash frac\{p\backslash alpha+q\}\{r\backslash alpha+s\}$ where $p,q,r,s\backslash in\backslash Z$ then $M(\backslash beta)\backslash ge\backslash frac\{M(\backslash alpha)\}\{\backslash left\backslash vert\; ps-rq\; \backslash right\backslash vert\}$.{{Cite journal|arxiv=1510.02407|title=Markov constant and quantum instabilities|journal=Journal of Physics A: Mathematical and Theoretical|volume=49|issue=15|pages=155201|doi=10.1088/1751-8113/49/15/155201|year=2016|last1=Pelantová|first1=Edita|last2=Starosta|first2=Štěpán|last3=Znojil|first3=Miloslav|bibcode=2016JPhA...49o5201P|s2cid=119161523 }} In particular if $\backslash left\backslash vert\; ps-rq\; \backslash right\backslash vert=1$ then $M(\backslash beta)=M(\backslash alpha)$.{{Cite book|title=Encyclopaedia of Mathematics|last=Hazewinkel|first=Michiel|publisher=Springer Science & Business Media|year=1990|isbn=9781556080050|pages=106}}

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

Burger et al. (2002){{Cite journal|last1=Burger|first1=Edward B.|last2=Folsom|first2=Amanda|last3=Pekker|first3=Alexander|last4=Roengpitya|first4=Rungporn|last5=Snyder|first5=Julia|date=2002|title=On a quantitative refinement of the Lagrange spectrum|journal=Acta Arithmetica|volume=102|issue=1|pages=59–60|bibcode=2002AcAri.102...55B|doi=10.4064/aa102-1-5|doi-access=free}} provides a formula for which the quadratic irrationality $\backslash alpha\_n$

whose Markov constant is the n^th Lagrange number:

$\backslash alpha\_n=\backslash frac\{2u-3m\_n+\backslash sqrt\{9m\_n^2-4\}\}\{2m\_n\}$ where $m\_n$ is the n^th Markov number, and {{Mvar|u}} is the smallest positive integer such that $m\_n\backslash mid\; u^2+1$.

Nicholls (1978){{Cite journal|last=Nicholls|first=Peter|date=1978|title=Diophantine Approximation via the Modular Group|journal=Journal of the London Mathematical Society |series=Second Series|volume=17|pages=11–17|doi=10.1112/jlms/s2-17.1.11}} provides a geometric proof of this (based on circles tangent to each other), providing a method that these numbers can be systematically found.

{{Image frame

|caption=A demonstration that {{math|{{sqrt|10}}/2}} has Markov constant {{math|{{sqrt|10}}}}, as stated in the example below. This plot graphs {{math|y(k)}} = {{math|{{sfrac|k²{{!}}α-{{sfrac|f(αk)|k}}{{!}}}}}} against {{math|log(k)}} (the natural log of k) where {{math|f(x)}} is the nearest integer to {{math|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 {{math|{{sqrt|10}}}} is approached.

|max-width=30%|content={{Markov constant chart}}

|width=360

|align=right}}

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

: $\backslash begin\{align\}\; M\backslash left\; (\; \backslash frac\{\backslash sqrt\{10\}\}\{2\}\; \backslash right\; )$

& = \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 E (mathematical constant)#Representations is unbounded.

Consider $n=6$; Then $m\_n=34$. By trial and error it can be found that $u=13$. Then

:

& = \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}

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

{{Reflist}}

Category:Continued fractions

Category:Diophantine approximation