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 Zap. Nauchn. Sem. POMI, 2018, Volume 469, Pages 32–63 (Mi znsl6605)

The karyon algorithm for decomposition into multidimensional continued fractions

V. G. Zhuravlevab

a Steklov Mathematical Institute of Russian Academy of Sciences, Moscow, Russia

Abstract: In this paper we propose a universal karyon algorithm, applicable to any set of real numbers $\alpha=(\alpha_1,…,\alpha_d)$, which is a modification of the simplex-karyon algorithm. The main difference is an infinite sequence $\mathbf T=\mathbf T_0,\mathbf T_1,…,\mathbf T_n,…$ of $d$-dimensional parallelohedra $\mathbf T_n$ instead of the simplex sequence. Each parallelohedron $\mathbf T_n$ is obtained from the previous $\mathbf T_{n-1}$ by means of the differentiation $\mathbf T_n=\mathbf T^{\sigma_n}_{n-1}$. Parallelohedra $\mathbf T_n$ represent itself karyons of certain induced toric tilings. A certain algorithm ($\varrho$-strategy) of the choice of infinite sequences $\sigma=\{\sigma_1,\sigma_2,…,\sigma_n,…\}$ of derivations $\sigma_n$ is specified. This algorithm provides the convergence $\varrho(\mathbf T_n)\to0$ if $n\to+\infty$, where $\varrho(\mathbf T_n)$ denotes the radius of the parallelohedron $\mathbf T_n$ in the metric $\varrho$ chosen as an objective function. It is proved that the parallelohedra $\mathbf T_n$ have the minimum property, i.e. the karyon approximation algorithm is the best with respect to karyon $\mathbf T_n$-norms. Also we get an estimate for the approximation rate of real numbers $\alpha=(\alpha_1,…,\alpha_d)$ by multidimensional continued fractions.

Key words and phrases: multidimensional continued fractions, the best approximations, simplex-karyon algorithm.

 Funding Agency Grant Number Russian Science Foundation 14-11-00433

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English version:
Journal of Mathematical Sciences (New York), 2019, 242:4, 487–508

Bibliographic databases:

UDC: 511.3

Citation: V. G. Zhuravlev, “The karyon algorithm for decomposition into multidimensional continued fractions”, Algebra and number theory. Part 1, Zap. Nauchn. Sem. POMI, 469, POMI, St. Petersburg, 2018, 32–63; J. Math. Sci. (N. Y.), 242:4 (2019), 487–508

Citation in format AMSBIB
\Bibitem{Zhu18} \by V.~G.~Zhuravlev \paper The karyon algorithm for decomposition into multidimensional continued fractions \inbook Algebra and number theory. Part~1 \serial Zap. Nauchn. Sem. POMI \yr 2018 \vol 469 \pages 32--63 \publ POMI \publaddr St.~Petersburg \mathnet{http://mi.mathnet.ru/znsl6605} \mathscinet{http://www.ams.org/mathscinet-getitem?mr=3885095} \transl \jour J. Math. Sci. (N. Y.) \yr 2019 \vol 242 \issue 4 \pages 487--508 \crossref{https://doi.org/10.1007/s10958-019-04492-7} \scopus{https://www.scopus.com/record/display.url?origin=inward&eid=2-s2.0-85072107119}