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2011 Markoff–Lagrange spectrum and extremal numbers
Damien Roy
Author Affiliations +
Acta Math. 206(2): 325-362 (2011). DOI: 10.1007/s11511-011-0065-z


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) = liminfq→∞q||qξ|| 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.

Funding Statement

Work partially supported by NSERC and CRM.


Download Citation

Damien Roy. "Markoff–Lagrange spectrum and extremal numbers." Acta Math. 206 (2) 325 - 362, 2011.


Received: 16 June 2009; Published: 2011
First available in Project Euclid: 31 January 2017

zbMATH: 1257.11071
MathSciNet: MR2810855
Digital Object Identifier: 10.1007/s11511-011-0065-z

Primary: 11J06
Secondary: 11J13

Rights: 2011 © Institut Mittag-Leffler


Vol.206 • No. 2 • 2011
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