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Diophantine approximation in \(\mathbb Q(\sqrt{-5})\) and \(\mathbb Q(\sqrt{-6})\). (English) Zbl 1091.11026

The author continues his successful program to extend Ford’s geometric approach to generalized Diophantine approximation, applying his earlier work on “Farey polytopes” (and generalized continued fractions) to the setting of Diophantine approximation by elements of a fixed complex field, here the fields of the title.
The geometric approach uses an appropriate discrete group of isometries of hyperbolic three-space to give the approximation. The analog of the classical Markov spectrum is given by the spectrum of closest approaches to infinity by geodesics on the quotient by the group. In each of the two cases treated, the author is able to reduce to the consideration of the spectrum for a certain maximal Fuchsian subgroup of the appropriate Bianchi group.
A fascinating new phenomenon is displayed: the existence of points in the spectrum for which there are infinitely many distinct geodesics achieving this value. Furthermore, the discrete part of the spectrum is determined in each case.

MSC:

11J06 Markov and Lagrange spectra and generalizations
11F06 Structure of modular groups and generalizations; arithmetic groups
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References:

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