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[edit]
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 when it is evaluated at non-zero vectors of the grid , 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:
Lagrange spectrum[edit]
Starting from Hurwitz's theorem on Diophantine approximation, that any real number has a sequence of rational approximations m/n tending to it with
it is possible to ask for each value of 1/c with 1/c ≥ √5 about the existence of some for which
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
where m is chosen as an integer function of n to make the difference minimal. This is a function of , and the reciprocal of the Lagrange spectrum is the range of values it takes on irrational numbers.
Relation with Markov spectrum[edit]
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
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[edit]
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]
Theorem — Given , the Hausdorff dimension of is equal to the Hausdorff dimension of . Moreover, if d is the function defined as , where dimH denotes the Hausdorff dimension, then d is continuous and maps R onto [0,1].
See also[edit]
References[edit]
- ^ Cassels (1957) p.18
- ^ Freiman's Constant Weisstein, Eric W. "Freiman's Constant." From MathWorld—A Wolfram Web Resource), accessed 26 August 2008
- ^ Cusick, Thomas; Flahive, Mary (1989). "The Markoff and Lagrange spectra compared". The Markoff and Lagrange Spectra. Mathematical Surveys and Monographs. Vol. 30. pp. 35–45. doi:10.1090/surv/030/03. ISBN 9780821815311.
- ^ Moreira, Carlos Gustavo T. De A. (July 2018). "Geometric properties of the Markov and Lagrange spectra". Annals of Mathematics. 188 (1): 145–170. arXiv:1612.05782. doi:10.4007/annals.2018.188.1.3. ISSN 0003-486X. JSTOR 10.4007/annals.2018.188.1.3. S2CID 15513612.
Further reading[edit]
- Aigner, Martin (2013). Markov's theorem and 100 years of the uniqueness conjecture : a mathematical journey from irrational numbers to perfect matchings. New York: Springer. ISBN 978-3-319-00887-5. OCLC 853659945.
- 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.
- Cassels, J.W.S. (1957). An introduction to Diophantine approximation. Cambridge Tracts in Mathematics and Mathematical Physics. Vol. 45. Cambridge University Press. Zbl 0077.04801.
External links[edit]
- "Markov spectrum problem", Encyclopedia of Mathematics, EMS Press, 2001 [1994]