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The karyon algorithm for decomposition into multidimensional continued fractions
Zapiski Nauchnykh Seminarov POMI, Algebra and number theory. Part 1, Tome 469 (2018), pp. 32-63 Cet article a éte moissonné depuis la source Math-Net.Ru

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In this paper we propose a universal karyon algorithm, applicable to any set of real numbers $\alpha=(\alpha_1,\dots,\alpha_d)$, which is a modification of the simplex-karyon algorithm. The main difference is an infinite sequence $\mathbf T=\mathbf T_0,\mathbf T_1,\dots,\mathbf T_n,\dots$ 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 $\sigma=\{\sigma_1,\sigma_2,\dots,\sigma_n,\dots\}$ 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 $\alpha=(\alpha_1,\dots,\alpha_d)$ by multidimensional continued fractions. 
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     author = {V. G. Zhuravlev},
     title = {The karyon algorithm for decomposition into multidimensional continued fractions},
     journal = {Zapiski Nauchnykh Seminarov POMI},
     pages = {32--63},
     year = {2018},
     volume = {469},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/ZNSL_2018_469_a1/}
}
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V. G. Zhuravlev. The karyon algorithm for decomposition into multidimensional continued fractions. Zapiski Nauchnykh Seminarov POMI, Algebra and number theory. Part 1, Tome 469 (2018), pp. 32-63. http://geodesic.mathdoc.fr/item/ZNSL_2018_469_a1/

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