Markov spectrum

In mathematics, the Markov spectrum devised by Andrey Markov is a complicated set of real numbers arising in the theory of diophantine approximation, and containing all the real numbers larger than Freiman's constant.^[1]

Context

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, 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.

The initial part of the Lagrange spectrum, namely the part lying in the interval [√5, 3), is associated with some binary quadratic forms that are indefinite (so factoring into two real linear forms). The first few values are √5, √8, (√221)/5, (√1517)/13, ... .^[2] The Markov spectrum deals directly with the phenomena associated to those quadratic forms.

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 OEIS).

Real numbers greater than F are also members of the Markov spectrum.^[3]

See also

• Markov number

References

1. Markov Spectrum Weisstein, Eric W. "Freiman's Constant." From MathWorld–A Wolfram Web Resource), accessed 26 Aug 2008
2. Cassels (1957) p.18
3. Freiman's Constant Weisstein, Eric W. "Freiman's Constant." From MathWorld—A Wolfram Web Resource), accessed 26 Aug 2008

• Cassels, J.W.S. (1957). An introduction to Diophantine approximation. Cambridge Tracts in Mathematics and Mathematical Physics 45. Cambridge University Press. Zbl 0077.04801.

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.

External links

• Hazewinkel, Michiel, ed. (2001), "Markov spectrum problem", Encyclopedia of Mathematics, Springer, ISBN 978-1-55608-010-4