Том 242, № 4 (2019)

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.

By the method of differentiation of induced toric tilings, periodic expansions for algebraic irrationalities in multidimensional continued fractions are found. These expansions give the best karyon approximations with respect to polyhedral norms. The above irrationalities are obtained by the composition of backward continued fraction mappings and unimodular transformations of algebraic units that are expanded in purely periodic continued fractions. Karyon expansions have several invariants: recurrence relations for the numerators and denominators of the convergents of continued fractions and the rate of multidimensional approximation of irrationalities by rational numbers.

The paper discusses the statement of the problem of constructing a big zeta function. This problem is related to an arithmetic Hurwitz formula. Two candidates for the part of the big zeta are suggested. Representations and ramification structures related to Kummer’s tower are studied.