Differentiation of Induced Toric Tiling and Multidimensional Approximations of Algebraic Numbers
- Авторы: Zhuravlev V.1
-
Учреждения:
- Vladimir State University
- Выпуск: Том 222, № 5 (2017)
- Страницы: 544-584
- Раздел: Article
- URL: https://journals.rcsi.science/1072-3374/article/view/239248
- DOI: https://doi.org/10.1007/s10958-017-3321-8
- ID: 239248
Цитировать
Аннотация
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 Krσ. They are used for approximation of 0 ∈ \( \mathbb{T} \)D by an infinite sequence of points xj ≡ jα mod ℤD, j = 0, 1, 2, . . . , where α = (α1, . . . , αD) is a vector whose coordinates α1, . . . , αD 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.
Об авторах
V. Zhuravlev
Vladimir State University
Автор, ответственный за переписку.
Email: vzhuravlev@mail.ru
Россия, Vladimir