Markov spectrum

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

Further information: Lagrange number

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 ≥ √5 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 √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

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

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 initial part of the Lagrange spectrum, namely the part lying in the interval [√5, 3), is also the initial part of Markov spectrum. The first few values are √5, √8, √221/5, √1517/13, ... 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:

$F=\frac{2221564096+283748\sqrt{462}}{491993569}=4.5278295661\dots$ (sequence A118472 in the OEIS).

All real numbers in [$F,\mathrm{\infty }$) - known as Hall’s ray - are members of the Lagrange spectrum. Moreover, it is possible to prove that L is strictly contained in M.

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:

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

See also

• Markov number

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.