In mathematics, the Markov spectrum devised by

Andrey Markov Andrey Andreyevich Markov, first name also spelled "Andrei", in older works also spelled Markoff) (14 June 1856 – 20 July 1922) was a Russian mathematician best known for his work on stochastic processes. A primary subject of his research lat ...

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 In mathematics, a quadratic form is a polynomial with terms all of degree two ("form" is another name for a homogeneous polynomial). For example, :4x^2 + 2xy - 3y^2 is a quadratic form in the variables and . The coefficients usually belong to a ...

given by ''f''(''x'',''y'') = ''ax''² + ''bxy'' + ''cy''² and suppose that its

discriminant In mathematics, the discriminant of a polynomial is a quantity that depends on the coefficients and allows deducing some properties of the roots without computing them. More precisely, it is a polynomial function of the coefficients of the origi ...

is fixed, say equal to −1/4. In other words, ''b''² − 4''ac'' = 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^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\

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\right , <\frac,

it is possible to ask for each value of 1/''c'' with 1/''c'' ≥ about the existence of some

\xi

for which :

\left , \xi-\frac\right , <\frac

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 (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 In mathematics, the limit inferior and limit superior of a sequence can be thought of as limiting (that is, eventual and extreme) bounds on the sequence. They can be thought of in a similar fashion for a function (see limit of a function). For a ...

. For that, consider :

\liminf_n^2\left , \xi-\frac\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 , is equal to the Markov spectrum. The first few values are , , /5, /13, ... 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

L_n = \sqrt.

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

F = \frac = 4.5278295661\dots

. Real numbers greater than ''F'' are also members of the Markov spectrum.Freiman's Constant
Weisstein, Eric W. "Freiman's Constant." From MathWorld—A Wolfram Web Resource), accessed 26 August 2008 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 Markov number

References

External links