Abstract:
Simplex-module algorithm (SM-algorithm) for expansion of algebraic numbers α=(α1,…,αd) in multidimensional continued fractions is offered. The method is based on 1) minimal rational simplices s, where α∈s, and 2) Pisot matrices Pα for which ˆα=(α1,…,αd,1) is eigenvector. A multi-dimensional generalization of the Lagrange theorem is proved.
Key words and phrases:
multidimensional continued fractions, best approximation, multidimensional generalization of Lagrange's theorem.
Citation:
V. G. Zhuravlev, “Simplex-module algorithm for expansion of algebraic numbers in multidimensional continued fractions”, Analytical theory of numbers and theory of functions. Part 32, Zap. Nauchn. Sem. POMI, 449, POMI, St. Petersburg, 2016, 130–167; J. Math. Sci. (N. Y.), 225:6 (2017), 924–949
\Bibitem{Zhu16}
\by V.~G.~Zhuravlev
\paper Simplex-module algorithm for expansion of algebraic numbers in multidimensional continued fractions
\inbook Analytical theory of numbers and theory of functions. Part~32
\serial Zap. Nauchn. Sem. POMI
\yr 2016
\vol 449
\pages 130--167
\publ POMI
\publaddr St.~Petersburg
\mathnet{http://mi.mathnet.ru/znsl6325}
\mathscinet{http://mathscinet.ams.org/mathscinet-getitem?mr=3580134}
\transl
\jour J. Math. Sci. (N. Y.)
\yr 2017
\vol 225
\issue 6
\pages 924--949
\crossref{https://doi.org/10.1007/s10958-017-3506-1}
\scopus{https://www.scopus.com/record/display.url?origin=inward&eid=2-s2.0-85027385763}
Linking options:
https://www.mathnet.ru/eng/znsl6325
https://www.mathnet.ru/eng/znsl/v449/p130
This publication is cited in the following 7 articles:
V. G. Zhuravlev, “L−-Algorithm for Approximation of Diophantine Systems of Linear Forms”, J Math Sci, 261:4 (2022), 517
V. G. Zhuravlev, “Symmetry Properties of Karyon Tilings”, J Math Sci, 264:2 (2022), 150
V. G. Zhuravlev, “Matrix-Fractional Invariance of Diophantine Systems of Linear Forms”, J Math Sci, 264:2 (2022), 103
V. G. Zhuravlev, “Diophantine Approximations of Linear Forms”, J Math Sci, 261:4 (2022), 503
V. G. Zhuravlev, “Local Structure of Karyon Tilings”, J Math Sci, 264:2 (2022), 122
V. G. Zhuravlev, “Lokalnyi algoritm postroeniya proizvodnykh razbienii dvumernogo tora”, Algebra i teoriya chisel. 2, Zap. nauchn. sem. POMI, 479, POMI, SPb., 2019, 85–120
V. G. Zhuravlev, “Simplex–karyon algorithm of multidimensional continued fraction expansion”, Proc. Steklov Inst. Math., 299 (2017), 268–287
Citation in format AMSBIB
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