Differentiation of Induced Toric Tiling and Multidimensional Approximations of Algebraic Numbers

Abstract

The paper considers the induced tilings \( \mathcal{T} \) = \( \mathcal{T} \) |_Kr of the D-dimensional torus \( \mathbb{T} \)^D generated by embedded karyons Kr. On \( \mathcal{T} \) , differentiation operations σ : \( \mathcal{T} \) −→\( \mathcal{T} \)^σ are defined, which provide the induced tilings \( \mathcal{T} \)^σ = \( \mathcal{T} \) |_Krσ of the same torus \( \mathbb{T} \)^D with the derivative karyon Kr^σ. They are used for approximation of 0 ∈ \( \mathbb{T} \)^D by an infinite sequence of points x_j ≡ jα mod ℤ^D, j = 0, 1, 2, . . . , where α = (α₁, . . . , α_D) is a vector whose coordinates α₁, . . . , α_D belong to an algebraic field ℚ(θ) of degree D+1 over the rational field ℚ. To this end, an infinite sequence of convex parallelohedra T ⁽ⁱ⁾ ⊂ \( \mathbb{T} \)^D, i = 0, 1, 2, . . ., is constructed, and natural orders m⁽⁰⁾ < m⁽¹⁾ < · · · < m⁽ⁱ⁾ < · · · for T ⁽ⁱ⁾ are defined. Then the above parallelohedra contain a subsequence of points \( {\left\{{x}_{j^{\prime }}\right\}}_{j^{\prime }=1}^{\infty } \) that are the best approximations of 0 ∈ \( \mathbb{T} \)^D. Bibliography: 25 titles.