Arithmetical and geometrical aspects of homogeneous Diophantine approximation by algebraic numbers in a given number field. (English) Zbl 1231.11072

Bugeaud, Yann (ed.) et al., Dynamical systems and Diophantine approximation. Paris: Société Mathématique de France (ISBN 978-2-85629-303-4/pbk). Séminaires et Congrès 19, 37-48 (2009).
In a first part, the author emphasizes the link between the Markoff and Lagrange spectra and shows the connection between minima of quadratic forms and homogeneous diophantine approximation in one variable. In a second part he shows some generalizations in terms of hyperbolic geometry by consideration of geodesic flow on Weyl chambers (see the author, Approximation diophantienne, dynamique des chambres de Weyl et répartition d’orbites de réseaux. Ph. D. Thesis, Université de Lille I (2002), http://tel.archives-ouvertes.fr/tel-00158036/fr/), and also recovers and generalizes some of the results of the first part.
We cho0se to detail the first part: Let \(\mathbb K\) be a number field of signature \((r,s)\), \(\mathcal O\) its ring of integers, \(C(\mathbb K)\) be the set of ideal classes and \(E=\mathbb R^r\times \mathbb C^s\). Let the norm \(N: E\rightarrow \mathbb R\) \[ x=(x_1,\dots,x_{r+s})\rightarrow \prod_{i=1}^r|x_i| \prod_{j=r+1}^{r+s} |x_j|^2. \]
– Markoff spectra: Let \(Q\) be an indefinite nondegenerate quadratic form \(Q(x,y)=ax^2+bxy+cy^2, \;a,b,c\in E\), so such that the discriminant \(\Delta(Q)=b^2-4ac\) has positive real components and non zero complex components. For any \(\mathcal I\in C(\mathbb K)\), let \((\mathcal O^2)_{\mathcal I}\) be the set of couples \((p,q)\in \mathcal O^2\backslash\{0,0\}\) such that the ideal \(\langle p,q\rangle\) generated by \(p\) and \(q\) belongs to \(\mathcal I\). The normalized infimum over \((\mathcal O^2)_{\mathcal I}\) is defined by \(\mu_{\mathcal I}(Q)=\inf _{(x,y)\in (\mathcal O^2)_{\mathcal I}}\frac{N(Q(x,y))}{N(\Delta(Q))^{1/2}},\) and the global normalized infimum \(\mu_\mathbb K(Q)= \inf_{\mathcal I\in C(\mathbb K)}\mu_{\mathcal I}(Q)\). The Markoff spectra \(M_{\mathcal I}\) and \(M_{\mathbb K}\) are defined as images of \(\mu_{\mathcal I}\), \(\mu_{\mathbb K}\) for all the indefinite nondegenerate quadratic forms.
– Lagrange spectra: For \(\mathcal I\in C(\mathbb K)\), let \(\mathbb K_{\mathcal I}=\{\frac{p}{q} \,|\, \langle p,q\rangle\, \in \mathcal I\}\). We define the approximation constant of \(x\in E-\mathbb K\) by \[ \nu_{\mathcal I}(x)=\Big(\lim\inf _{p/q\in \mathbb K_{\mathcal I}, \;p/q\rightarrow x}N(q)^2N(x-p/q)\Big)^{1/{[\mathbb K:\mathbb Q]}}. \] See similar definitions in [E. B. Burger, Pac. J. Math. 152, No. 2, 211–254 (1992; Zbl 0725.11032)] and [R. Quême, Journées arithmétiques, Exp. Congr., Luminy/Fr. 1989, Astérisque 198-200, 273–283 (1991; Zbl 0756.11016)] for some Dirichlet type theorems. The Lagrange spectra \( L_{\mathcal I}\) and \(L_{\mathbb K}\) are the images of \(\nu_{\mathcal I}\) and \(\nu_{\mathbb K}\). The author proves:
For all \(\mathcal I\in C(\mathbb K)\), we have \(L_{\mathcal I}\subset M_{\mathcal I}\) and both sets are bounded. We have also \(L_{\mathbb K}\subset M_{\mathbb K}\).
The sets \(M_{\mathcal I}\) are closed, and so is \(M_{\mathbb K}\).
We have \(\sup L_{\mathbb K}=\sup M_{\mathbb K}\) and both suprema are in fact maxima.
In the case of imaginary quadratic fields the author proves in [F. Maucourant, Ergodic Theory Dyn. Syst. 23, No. 1, 193–205 (2003; Zbl 1049.11068)] an analogue of a Cusick [T. W. Cusick and M. E. Flahive, The Markoff and Lagrange spectra. Providence, RI: AMS (1989; Zbl 0685.10023)] result for \(\mathbb Q\):
Theorem. Assume that \(\mathbb K\) is an imaginary quadratic field. Let \(\mathcal I\in C(\mathbb K)\) and denote by \(\mathcal Q\) the set of quadratic numbers over \(\mathbb K\). We have \[ L_{\mathcal I}= \overline{\{\nu_{\mathcal I}:\;x \in\mathcal Q\}}, \] and in particular it is a closed set.
In the last section, the author observes that a Furstenberg-Margulis conjecture (see [D. Ferte, Dynamique topologique d’une action de groupe sur un espace homogène: exemple d’actions unipotentes et diagonales. Ph. D. thesis Université Rennes I (2003), http://tel.archives-ouvertes.fr/tel-00007213/fr/]) would imply the following conjecture:
Every indefinite nondegenerate quadratic form \(Q\) with coefficients in \(\mathbb R^2\) such that \(\mu_\mathbb K(Q)>0\) is proportional to a quadratic form with coefficients in \(K\), and that, for rank \(r+s\geq 2\), the Markoff and Lagrange spectra would be sequences tending to \(0\).
For the entire collection see [Zbl 1213.11004].


11J06 Markov and Lagrange spectra and generalizations
11J17 Approximation by numbers from a fixed field
37A17 Homogeneous flows