{{Short description|Complicated set of real numbers}} In mathematics, the '''Markov spectrum''', devised by Andrey Markov, is a complicated set of real numbers arising in Markov Diophantine equations and also in the theory of Diophantine approximation.

== Quadratic form characterization == Consider a quadratic form given by ''f''(''x'',''y'') = ''ax''<sup>2</sup> + ''bxy'' + ''cy''<sup>2</sup> and suppose that its discriminant is fixed, say equal to −1/4. In other words, ''b''<sup>2</sup>&nbsp;−&nbsp;4''ac'' =&nbsp;1.

One can ask for the minimal value achieved by <math> \left\vert f(x,y) \right\vert </math> when it is evaluated at non-zero vectors of the grid <math>\mathbb{Z}^2</math>, and if this minimum does not exist, for the infimum.

The Markov spectrum ''M'' is the set obtained by repeating this search with different quadratic forms with discriminant fixed to −1/4:<math display="block">M = \left\{ \left(\inf_{(x,y)\in \Z^2 \smallsetminus \{(0,0)\}} |f(x,y)| \right)^{-1} : f(x,y) = ax^2 + bxy + cy^2,\ b^2- 4ac = 1 \right\}</math>

==Lagrange spectrum== {{details|Lagrange number}} Starting from Hurwitz's theorem on Diophantine approximation, that any real number <math>\xi</math> has a sequence of rational approximations ''m''/''n'' tending to it with

:<math>\left |\xi-\frac{m}{n}\right |<\frac{1}{\sqrt{5}\, n^2},</math>

it is possible to ask for each value of 1/''c'' with 1/''c'' &ge; {{radic|5}} about the existence of some <math>\xi</math> for which

:<math>\left |\xi-\frac{m}{n}\right |<\frac{c} {n^2}</math>

for such a sequence, for which ''c'' is the best possible (maximal) value. Such 1/''c'' make up the '''Lagrange spectrum''' ''L'', a set of real numbers at least {{radic|5}} (which is the smallest value of the spectrum). The formulation with the reciprocal is awkward, but the traditional definition invites it; looking at the set of ''c'' instead allows a definition instead by means of an inferior limit. For that, consider :<math>\liminf_{n \to \infty}n^2\left |\xi-\frac{m}{n}\right |,</math>

where ''m'' is chosen as an integer function of ''n'' to make the difference minimal. This is a function of <math>\xi</math>, and the reciprocal of the Lagrange spectrum is the range of values it takes on irrational numbers.

=== Relation with Markov spectrum === The Lagrange spectrum is a proper subset of the Markov spectrum.<ref>{{Cite book|chapter=The Markoff and Lagrange spectra compared|last1=Cusick|first1=Thomas|last2=Flahive|first2=Mary|author2-link= Mary Flahive |pages=35–45|doi=10.1090/surv/030/03|title = The Markoff and Lagrange Spectra|volume = 30|series = Mathematical Surveys and Monographs|year = 1989|isbn = 9780821815311}}</ref> The initial part of the Lagrange spectrum, namely the part lying in the interval {{closed-open|{{radic|5}}, 3}}, is also the initial part of Markov spectrum. The first few values are {{radic|5}}, {{radic|8}}, {{radic|221}}/5, {{radic|1517}}/13, ...<ref>Cassels (1957) p.18</ref> and the ''n''th number of this sequence (that is, the ''n''th Lagrange number) can be calculated from the ''n''th Markov number by the formula<math display="block">L_n = \sqrt{9 - {4 \over {m_n}^2}}.</math>'''Freiman's constant''' is the name given to the end of the last gap in the Lagrange spectrum, namely:

: <math> F = \frac{2\,221\,564\,096 + 283\,748\sqrt{462}}{491\, 993\, 569} = 4.5278295661\dots</math> {{OEIS|A118472}}.

All real numbers in {{closed-open|<math>{{F}}, \infty</math>}} - known as Hall’s ray - are members of the Lagrange spectrum.<ref name=mathworld2>[http://mathworld.wolfram.com/FreimansConstant.html Freiman's Constant] Weisstein, Eric W. "Freiman's Constant." From MathWorld&mdash;A Wolfram Web Resource), accessed 26 August 2008</ref>

== Geometry of Markov and Lagrange spectrum == On one hand, the initial part of the Markov and Lagrange spectrum lying in the interval [{{radic|5}}, 3) are both equal and they are a discrete set. On the other hand, the final part of these sets lying after Freiman's constant are also equal, but a continuous set. The geometry of the part between the initial part and final part has a fractal structure, and can be seen as a geometric transition between the discrete initial part and the continuous final part. This is stated precisely in the next theorem:<ref>{{Cite journal|last=Moreira |first=Carlos Gustavo|date=July 2018|title=Geometric properties of the Markov and Lagrange spectra |journal=Annals of Mathematics|volume=188|issue=1| pages=145–170 |doi=10.4007/annals.2018.188.1.3 |issn=0003-486X | arxiv=1612.05782| jstor=10.4007/annals.2018.188.1.3|s2cid=15513612 }}</ref>{{math theorem|Given <math>t \in \R</math>, the Hausdorff dimension of <math>L\cap(-\infty,t)</math> is equal to the Hausdorff dimension of <math>M\cap(-\infty,t)</math>. Moreover, if ''d'' is the function defined as <math>d(t):=\dim_{H}(M\cap(-\infty,t))</math>, where dim<sub>''H''</sub> denotes the Hausdorff dimension, then ''d'' is continuous and maps '''R''' onto [0,1].}}

==See also== *Markov number

==References== {{reflist}}

==Further reading== * {{cite book | last=Aigner | first=Martin | author-link=Martin Aigner | title=Markov's theorem and 100 years of the uniqueness conjecture : a mathematical journey from irrational numbers to perfect matchings | publisher=Springer | publication-place= New York | year=2013 | isbn=978-3-319-00887-5 | oclc=853659945}} *Conway, J. H. and Guy, R. K. The Book of Numbers. New York: Springer-Verlag, pp.&nbsp;188&ndash;189, 1996. *Cusick, T. W. and Flahive, M. E. The Markov and Lagrange Spectra. Providence, RI: Amer. Math. Soc., 1989. * {{cite book | first=J.W.S. | last=Cassels | authorlink=J. W. S. Cassels | title=An introduction to Diophantine approximation | series=Cambridge Tracts in Mathematics and Mathematical Physics | volume=45 | publisher=Cambridge University Press | year=1957 | zbl=0077.04801 }}

==External links== *{{Springer|id=m/m062540|title=Markov spectrum problem}}

Category:Diophantine approximation Category:Quadratic forms Category:Combinatorics