# MathSciDoc: An Archive for Mathematician ∫

#### Spectral Theory and Operator Algebramathscidoc:1701.32001

Acta Mathematica, 206, (2), 325-362, 2009.6
Let $\gamma = \frac{1}{2}\left( {1 + \sqrt {5} } \right)$ denote the golden ratio. H. Davenport and W. M. Schmidt showed in 1969 that, for each non-quadratic irrational real number ξ, there exists a constant$c$> 0 with the property that, for arbitrarily large values of$X$, the inequalities $\left| {{x_0}} \right| \leqslant X,\,\,\,\left| {{x_0}\xi - {x_1}} \right| \leqslant c{X^{{{{ - 1}} \left/ {\gamma } \right.}}}\,\,\,{\text{and}}\,\,\,\left| {{x_0}{\xi^2} - {x_2}} \right| \leqslant c{X^{{{{ - 1}} \left/ {\gamma } \right.}}}$ admit no non-zero solution $\left( {{x_0},{x_1},{x_2}} \right) \in {\mathbb{Z}^3}$ . Their result is best possible in the sense that, conversely, there are countably many non-quadratic irrational real numbers ξ such that, for a larger value of c, the same inequalities admit a non-zero integer solution for each$X$≥ 1. Such$extremal$numbers are transcendental and their set is stable under the action of ${\text{G}}{{\text{L}}_2}\left( \mathbb{Z} \right)$ on $\mathbb{R}\backslash \mathbb{Q}$ by linear fractional transformations. In this paper, it is shown that there exist extremal numbers ξ for which the Lagrange constant ν(ξ) = liminf_{$q$→∞}$q$||$q$ξ|| is $\frac{1}{3}$ , the largest possible value for a non-quadratic number, and that there is a natural bijection between the ${\text{G}}{{\text{L}}_2}\left( \mathbb{Z} \right)$ -equivalence classes of such numbers and the non-trivial solutions of Markoff’s equation.
Lagrange spectrum; Markoff spectrum; simultaneous Diophantine approximation; extremal numbers
@inproceedings{damien2009markoff–lagrange,
title={Markoff–Lagrange spectrum and extremal numbers},
author={Damien Roy},
url={http://archive.ymsc.tsinghua.edu.cn/pacm_paperurl/20170108203356784532741},
booktitle={Acta Mathematica},
volume={206},
number={2},
pages={325-362},
year={2009},
}

Damien Roy. Markoff–Lagrange spectrum and extremal numbers. 2009. Vol. 206. In Acta Mathematica. pp.325-362. http://archive.ymsc.tsinghua.edu.cn/pacm_paperurl/20170108203356784532741.