Markov spectrum

Template:Short description 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) = 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 $|f(x,y)|$ when it is evaluated at non-zero vectors of the grid ${\mathbb{Z}}^2$, 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:$$M=\{{(\underset{(x,y)\in {\mathbb{Z}}^2\setminus \{(0,0)\}}{inf}|f(x,y)|)}^{-1}:f(x,y)=a{x}^2+bxy+c{y}^2,b^2-4ac=1\}$$

Lagrange spectrum

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

$|\xi -\frac{m}{n}|<\frac{1}{\sqrt{5}{n}^2},$

it is possible to ask for each value of 1/c with 1/c ≥ Template:Radic about the existence of some $\xi$ for which

$|\xi -\frac{m}{n}|<\frac{c}{n^2}$

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 Template:Radic (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

$\underset{n\to \mathrm{\infty }}{lim\; inf}{n}^2|\xi -\frac{m}{n}|,$

where m is chosen as an integer function of n to make the difference minimal. This is a function of $\xi$, 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. The initial part of the Lagrange spectrum, namely the part lying in the interval Template:Closed-open, is also the initial part of Markov spectrum. The first few values are Template:Radic, Template:Radic, Template:Radic/5, Template:Radic/13, ... and the nth number of this sequence (that is, the nth Lagrange number) can be calculated from the nth Markov number by the formula$$L_n=\sqrt{9-\frac{4}{{m_n}^2}}.$$Freiman's constant is the name given to the end of the last gap in the Lagrange spectrum, namely:

$F=\frac{2221564096+283748\sqrt{462}}{491993569}=4.5278295661\dots$ Template:OEIS.

All real numbers in Template:Closed-open - known as Hall’s ray - are members of the Lagrange spectrum.

Geometry of Markov and Lagrange spectrum

On one hand, the initial part of the Markov and Lagrange spectrum lying in the interval [[[:Template:Radic]], 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:Template:Math theorem

See also

• Markov number

Further reading

• Template:Cite book
• 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.
• Template:Cite book

External links

• Template:Springer