Nonarchimedean quadratic Lagrange spectra and continued fractions in power series fields. (English) Zbl 1465.11163
The present paper deals with the nonarchimedean quadratic Lagrange spectrum defined by J. Parkkonen and F. Paulin [Math. Z. 294, No. 3–4, 1065–1084 (2020; Zbl 1461.11104)] and with continued fractions in power series fields.
A survey of this research is devoted to the following notions: the Lagrange spectrum, Hall’s ray, the quadratic Lagrange spectrum, Diophantine approximations, the complexity, the quadratic approximation constant, the (quadratic) Hurwitz constant, quadratic power series, etc. The special attention is given to continued fractions in power series fields.
The author gives the following description of the present research:
“Let \(\mathbb F_q\) be a finite field of order a positive power \(q\) of a prime number. We study the nonarchimedean quadratic Lagrange spectrum defined by Parkkonen and Paulin [loc. cit.] by considering the approximation by elements of the orbit of a given quadratic power series in \(\mathbb F_q((Y^{-1}))\), for the action by homographies and anti-homographies of \(\mathrm{PGL}_2(\mathbb F_q[Y ])\) on \(\mathbb F_q((Y^{-1})) \cup \{\infty\}\). While Parkkonen and Paulin’s approach used geometric methods of group actions on Bruhat-Tits trees, ours is based on the theory of continued fractions in power series fields.”
Connections between the present results and known results of other authors are discussed.
A survey of this research is devoted to the following notions: the Lagrange spectrum, Hall’s ray, the quadratic Lagrange spectrum, Diophantine approximations, the complexity, the quadratic approximation constant, the (quadratic) Hurwitz constant, quadratic power series, etc. The special attention is given to continued fractions in power series fields.
The author gives the following description of the present research:
“Let \(\mathbb F_q\) be a finite field of order a positive power \(q\) of a prime number. We study the nonarchimedean quadratic Lagrange spectrum defined by Parkkonen and Paulin [loc. cit.] by considering the approximation by elements of the orbit of a given quadratic power series in \(\mathbb F_q((Y^{-1}))\), for the action by homographies and anti-homographies of \(\mathrm{PGL}_2(\mathbb F_q[Y ])\) on \(\mathbb F_q((Y^{-1})) \cup \{\infty\}\). While Parkkonen and Paulin’s approach used geometric methods of group actions on Bruhat-Tits trees, ours is based on the theory of continued fractions in power series fields.”
Connections between the present results and known results of other authors are discussed.
Reviewer: Symon Serbenyuk (Kyïv)
MSC:
| 11J06 | Markov and Lagrange spectra and generalizations |
| 11J61 | Approximation in non-Archimedean valuations |
| 11J70 | Continued fractions and generalizations |
| 11R11 | Quadratic extensions |
Citations:
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