# 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.

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| when it is evaluated at non-zero vectors of the grid $\displaystyle{ \mathbb{Z}^2 }$, 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:$\displaystyle{ M = \left\{ \left(\inf_{(x,y)\in \mathbb{Z}^2\smallsetminus\{(0,0)\}} |f(x,y)| \right)^{-1} : f(x,y) = ax^2 + bxy + cy^2,\ b^2- 4ac = 1 \right\} }$

## Lagrange spectrum

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

$\displaystyle{ \left |\xi-\frac{m}{n}\right |\lt \frac{1}{\sqrt{5}\, n^2}, }$

it is possible to ask for each value of 1/c with 1/c5 about the existence of some $\displaystyle{ \xi }$ for which

$\displaystyle{ \left |\xi-\frac{m}{n}\right |\lt \frac{c} {n^2} }$

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

$\displaystyle{ \liminf_{n \to \infty}n^2\left |\xi-\frac{m}{n}\right |, }$

where m is chosen as an integer function of n to make the difference minimal. This is a function of $\displaystyle{ \xi }$, 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 equal to the Markov spectrum. The first few values are 5, 8, 221/5, 1517/13, ...[1] and the nth number of this sequence (that is, the nth Lagrange number) can be calculated from the nth Markov number by the formula

$\displaystyle{ L_n = \sqrt{9 - {4 \over {m_n}^2}}. }$

Freiman's constant is the name given to the end of the last gap in the Lagrange spectrum, namely:

$\displaystyle{ F = \frac{2\,221\,564\,096 + 283\,748\sqrt{462}}{491\, 993\, 569} = 4.5278295661\dots }$ (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 $\displaystyle{ t \in \mathbb{R} }$, the Hausdorff dimension of $\displaystyle{ L\cap(-\infty,t) }$ is equal to the Hausdorff dimension of $\displaystyle{ M\cap(-\infty,t) }$. Moreover, if d is the function defined as $\displaystyle{ d(t):=\dim_{H}(M\cap(-\infty,t)) }$, where dimH denotes the Hausdorff dimension, then d is continuous and maps R onto [0,1].