Abstract:
The notion of equivalence of multidimensional continued fractions is introduced. We consider some properties and state some conjectures related to the structure of the family of equivalence classes of two-dimensional periodic
continued fractions. Our approach to the study of the family of equivalence classes of two-dimensional periodic continued fractions leads to revealing special subfamilies of continued fractions for which the triangulations of the
torus (i.e., the combinatorics of their fundamental domains) are subjected to clear rules. Some of these subfamilies are studied in detail; the way to construct other similar subfamilies is indicated.
Keywords:
multidimensional continued fractions, convex hulls, integer operators, cubic extensions of Q.
Citation:
O. N. Karpenkov, “On Tori Triangulations Associated with Two-Dimensional Continued Fractions of Cubic Irrationalities”, Funktsional. Anal. i Prilozhen., 38:2 (2004), 28–37; Funct. Anal. Appl., 38:2 (2004), 102–110
\Bibitem{Kar04}
\by O.~N.~Karpenkov
\paper On Tori Triangulations Associated with Two-Dimensional Continued Fractions of Cubic Irrationalities
\jour Funktsional. Anal. i Prilozhen.
\yr 2004
\vol 38
\issue 2
\pages 28--37
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\transl
\jour Funct. Anal. Appl.
\yr 2004
\vol 38
\issue 2
\pages 102--110
\crossref{https://doi.org/10.1023/B:FAIA.0000034040.08573.22}
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Linking options:
https://www.mathnet.ru/eng/faa105
https://doi.org/10.4213/faa105
https://www.mathnet.ru/eng/faa/v38/i2/p28
This publication is cited in the following 17 articles:
A. A. Lodkin, “Klein Sail and Diophantine Approximation of a Vector”, J Math Sci, 247:5 (2020), 680
A. A. Lodkin, “Parus Kleina i diofantovy priblizheniya vektora”, Teoriya predstavlenii, dinamicheskie sistemy, kombinatornye i algoritmicheskie metody. XXX, Zap. nauchn. sem. POMI, 481, POMI, SPb., 2019, 63–73
A. A. Illarionov, “Some properties of three-dimensional Klein polyhedra”, Sb. Math., 206:4 (2015), 510–539
Karpenkov O., “Multidimensional Gauss Reduction Theory for Conjugacy Classes of Sl(N, Z)”, J. Theor. Nr. Bordx., 25:1 (2013), 99–109
Oleg Karpenkov, Algorithms and Computation in Mathematics, 26, Geometry of Continued Fractions, 2013, 301
Oleg Karpenkov, Algorithms and Computation in Mathematics, 26, Geometry of Continued Fractions, 2013, 249
Oleg Karpenkov, Algorithms and Computation in Mathematics, 26, Geometry of Continued Fractions, 2013, 281
Karpenkov O., “Continued fractions and the second Kepler law”, Manuscripta Math, 134:1–2 (2011), 157–169
O. N. Karpenkov, “Determination of Periods of Geometric Continued Fractions for Two-Dimensional Algebraic Hyperbolic Operators”, Math. Notes, 88:1 (2010), 28–38
Karpenkov O.N., Vershik A.M., “Rational approximation of maximal commutative subgroups of GL(n, R)”, J Fixed Point Theory Appl, 7:1 (2010), 241–263
Karpenkov O.N., “Constructing multidimensional periodic continued fractions in the sense of Klein”, Math. Comp., 78:267 (2009), 1687–1711
O. N. Karpenkov, “On an Invariant Möbius Measure and the Gauss–Kuzmin Face Distribution”, Proc. Steklov Inst. Math., 258 (2007), 74–86
Karpenkov O.N., “Completely empty pyramids on integer lattices and two-dimensional faces of multidimensional continued fractions”, Monatsh. Math., 152:3 (2007), 217–249
Karpenkov O., “Three examples of three-dimensional continued fractions in the sense of Klein”, C. R. Math. Acad. Sci. Paris, 343:1 (2006), 5–7
O. N. Karpenkov, “Classification of three-dimensional multistorey completely empty convex marked pyramids”, Russian Math. Surveys, 60:1 (2005), 165–166
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