# Markov spectrum

In mathematics, the **Markov spectrum** devised by Andrey Markov is a complicated set of real numbers arising in Markov Diophantine equation and also in the theory of Diophantine approximation.

## Quadratic form characterization

Consider a quadratic form given by *f*(*x*,*y*) = *ax*^{2} + *bxy* + *cy*^{2} and suppose that its discriminant is fixed, say equal to −1/4. In other words, *b*^{2} − 4*ac* = 1.

One can ask for the minimal value achieved by *|f|* when it is evaluated at non-zero vectors of the grid [math]\displaystyle{ \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]\displaystyle{ M = \left\{ \left(\inf_{(x,y)\in \mathbb{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

Starting from Hurwitz's theorem on Diophantine approximation, that any real number [math]\displaystyle{ \xi }[/math] has a sequence of rational approximations *m*/*n* tending to it with

- [math]\displaystyle{ \left |\xi-\frac{m}{n}\right |\lt \frac{1}{\sqrt{5}\, n^2}, }[/math]

it is possible to ask for each value of 1/*c* with 1/*c* ≥ √5 about the existence of some [math]\displaystyle{ \xi }[/math] for which

- [math]\displaystyle{ \left |\xi-\frac{m}{n}\right |\lt \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 √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]\displaystyle{ \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]\displaystyle{ \xi }[/math], and the reciprocal of the Lagrange spectrum is the range of values it takes on irrational numbers.

### Relation with Markov spectrum

The initial part of the Lagrange spectrum, namely the part lying in the interval [√5, 3), is equal to the Markov spectrum. The first few values are √5, √8, √221/5, √1517/13, ...^{[1]} 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]\displaystyle{ 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]\displaystyle{ F = \frac{2\,221\,564\,096 + 283\,748\sqrt{462}}{491\, 993\, 569} = 4.5278295661\dots }[/math] (sequence A118472 in the OEIS).

Real numbers greater than *F* are also members of the Markov spectrum.^{[2]} Moreover, it is possible to prove that *L* is strictly contained in *M*.^{[3]}

## Geometry of Markov and Lagrange spectrum

On one hand, the initial part of the Markov and Lagrange spectrum lying in the interval [√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:^{[4]}

Given [math]\displaystyle{ t \in \mathbb{R} }[/math], the Hausdorff dimension of [math]\displaystyle{ L\cap(-\infty,t) }[/math] is equal to the Hausdorff dimension of [math]\displaystyle{ M\cap(-\infty,t) }[/math]. Moreover, if

dis the function defined as [math]\displaystyle{ d(t):=\dim_{H}(M\cap(-\infty,t)) }[/math], where dim_{H}denotes the Hausdorff dimension, thendis continuous and mapsRonto [0,1].

## See also

## References

- ↑ Cassels (1957) p.18
- ↑ Freiman's Constant Weisstein, Eric W. "Freiman's Constant." From MathWorld—A Wolfram Web Resource), accessed 26 August 2008
- ↑ Cusick, Thomas; Flahive, Mary (1989). "The Markoff and Lagrange spectra compared".
*The Markoff and Lagrange Spectra*. Mathematical Surveys and Monographs.**30**. pp. 35–45. doi:10.1090/surv/030/03. ISBN 9780821815311. - ↑ Moreira, Carlos Gustavo T. De A. (July 2018). "Geometric properties of the Markov and Lagrange spectra".
*Annals of Mathematics***188**(1): 145–170. doi:10.4007/annals.2018.188.1.3. ISSN 0003-486X.

## Further reading

- 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.
- Cassels, J.W.S. (1957).
*An introduction to Diophantine approximation*. Cambridge Tracts in Mathematics and Mathematical Physics.**45**.*Cambridge University Press*.

## External links

- Hazewinkel, Michiel, ed. (2001), "Markov spectrum problem",
*Encyclopedia of Mathematics*, Springer Science+Business Media B.V. / Kluwer Academic Publishers, ISBN 978-1-55608-010-4, https://www.encyclopediaofmath.org/index.php?title=m/m062540

Original source: https://en.wikipedia.org/wiki/ Markov spectrum.
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