Markov spectrum

Consider a quadratic form given by f(x,y) = ax² + bxy + cy² and suppose that its discriminant is fixed, say equal to −1/4. In other words, b² − 4ac = 1.

One can ask for the minimal value achieved by $\backslash left\backslash vert\; f(x,y)\; \backslash right\backslash vert$ when it is evaluated at non-zero vectors of the grid $\backslash mathbb\{Z\}^2$, 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:$$M\; =\; \backslash left\backslash \{\; \backslash left(\backslash inf\_\{(x,y)\backslash in\; \backslash Z^2\; \backslash smallsetminus\; \backslash \{(0,0)\backslash \}\}\; |f(x,y)|\; \backslash right)^\{-1\}\; :\; f(x,y)\; =\; ax^2\; +\; bxy\; +\; cy^2,\backslash \; b^2-\; 4ac\; =\; 1\; \backslash right\backslash \}$$

{{details|Lagrange number}}

Starting from Hurwitz's theorem on Diophantine approximation, that any real number $\backslash xi$ has a sequence of rational approximations m/n tending to it with

:

it is possible to ask for each value of 1/c with 1/c ≥ {{radic|5}} about the existence of some $\backslash xi$ for which

:

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

:$\backslash liminf\_\{n\; \backslash to\; \backslash infty\}n^2\backslash left\; |\backslash xi-\backslash frac\{m\}\{n\}\backslash right\; |,$

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

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, ...Cassels (1957) p.18 and the nth number of this sequence (that is, the nth Lagrange number) can be calculated from the nth Markov number by the formula$$L\_n\; =\; \backslash sqrt\{9\; -\; \{4\; \backslash over\; \{m\_n\}^2\}\}.$$Freiman's constant is the name given to the end of the last gap in the Lagrange spectrum, namely:

: $F\; =\; \backslash frac\{2\backslash ,221\backslash ,564\backslash ,096\; +\; 283\backslash ,748\backslash sqrt\{462\}\}\{491\backslash ,\; 993\backslash ,\; 569\}\; =\; 4.5278295661\backslash dots$ {{OEIS|A118472}}.

All real numbers in {{closed-open|$\{\{F\}\},\; \backslash infty$}} - known as Hall’s ray - are members of the Lagrange spectrum.[http://mathworld.wolfram.com/FreimansConstant.html Freiman's Constant] Weisstein, Eric W. "Freiman's Constant." From MathWorld—A Wolfram Web Resource), accessed 26 August 2008 Moreover, it is possible to prove that L is strictly contained in M.{{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}}

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:{{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 }}{{math theorem|Given $t\; \backslash in\; \backslash R$, the Hausdorff dimension of $L\backslash cap(-\backslash infty,t)$ is equal to the Hausdorff dimension of $M\backslash cap(-\backslash infty,t)$. Moreover, if d is the function defined as $d(t):=\backslash dim\_\{H\}(M\backslash cap(-\backslash infty,t))$, where dim_H denotes the Hausdorff dimension, then d is continuous and maps R onto [0,1].}}

• Markov number

{{reflist}}

• {{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. 188–189, 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 }}

• {{Springer|id=m/m062540|title=Markov spectrum problem}}

Category:Diophantine approximation

Category:Quadratic forms

Category:Combinatorics