The Karyon Algorithm for Expansion in Multidimensional Continued Fractions

Abstract

The paper presents a universal karyon algorithm, applicable to an arbitrary collection of reals α = (α₁, . . . , α_d), which is a modification of the simplex-karyon algorithm. The main distinction is that instead of a simplex sequence, an infinite sequence T = T₀,T₁, . . . ,T_n, . . . of d-dimensional parallelohedra T_n appear. Every parallelohedron T_n is obtained from the previous one T_n−1 by differentiation,\( {\mathbf{T}}_n={\mathbf{T}}_{n-1}^{\sigma n} \). The parallelohedra T_n are the karyons of some induced toric tilings. A certain algorithm (ϱ-strategy) for choosing infinite sequences σ|={σ₁, σ₂, …, σ_n, …} of differentiations σ_n is specified. This algorithm ensures the convergence ϱ(T_n) −→ 0 as n → +∞, where ϱ(T_n) denotes the radius of the parallelohedron T_n in the metric ϱ chosen as the objective function. It is proved that the parallelohedra T_n have the minimality property, i.e., the karyon approximation algorithm is the best one with respect to the karyon T_n-norms. Also an estimate for the rate of approximation of real numbers α = (α₁, . . . , α_d) by multidimensional continued fractions is derived.