The karyon algorithm for decomposition into multidimensional continued fractions

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.