# 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) = ax$^2$ + bxy + cy$^2$ and suppose that its discriminant is fixed, say equal to −1/4. In other words, b$^2$ − 4ac = 1.
One can ask for the minimal value achieved by $\left\vert f(x,y)\right\vert$ 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:$M=\left\{\left(\inf _{(x,y)\in \mathbb {Z} ^{2}\smallsetminus \{(0,0)\}}|f(x,y)|\right)^{-1}:f(x,y)=ax^{2}+bxy+cy^{2},\ b^{2}-4ac=1\right\}$

## Lagrange spectrum

Starting from Hurwitz's theorem on Diophantine approximation, that any real number $\xi$ has a sequence of rational approximations m/n tending to it with
    $\left|\xi -{\frac {m}{n}}\right|<{\frac {1}{{\sqrt {5}}\,n^{2}}},$
it is possible to ask for each value of 1/c with 1/c ≥ √5 about the existence of some $\xi$ for which
    $\left|\xi -{\frac {m}{n}}\right|<{\frac {c}{n^{2}}}$
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
    $\liminf _{n\to \infty }n^{2}\left|\xi -{\frac {m}{n}}\right|,$
where m is chosen as an integer function of n to make the difference minimal. This is a function of $\xi$, 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$L_{n}={\sqrt {9-{4 \over {m_{n}}^{2}}}}.$Freiman's constant is the name given to the end of the last gap in the Lagrange spectrum, namely:
    $F={\frac {2\,221\,564\,096+283\,748{\sqrt {462}}}{491\,993\,569}}=4.5278295661\dots$ (sequence A118472 in the OEIS).
All real numbers in [${F},\infty$) - 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 $t\in \mathbb {R}$, the Hausdorff dimension of $L\cap (-\infty ,t)$ is equal to the Hausdorff dimension of $M\cap (-\infty ,t)$. Moreover, if d is the function defined as $d(t):=\dim _{H}(M\cap (-\infty ,t))$, where dim$_H$ denotes the Hausdorff dimension, then d is continuous and maps R onto [0,1].