The Best Approximation of Algebraic Numbers by Multidimensional Continued Fractions
| Autor: | V. G. Zhuravlev |
|---|---|
| Rok vydání: | 2020 |
| Předmět: |
Statistics and Probability
Rational number Constant coefficients Recurrence relation Degree (graph theory) Applied Mathematics General Mathematics 010102 general mathematics Common denominator 01 natural sciences 010305 fluids & plasmas Moduli Combinatorics 0103 physical sciences 0101 mathematics Algebraic number Unit (ring theory) Mathematics |
| Zdroj: | Journal of Mathematical Sciences. 249:32-53 |
| ISSN: | 1573-8795 1072-3374 |
| DOI: | 10.1007/s10958-020-04918-7 |
| Popis: | The karyon-module ( $$ \mathcal{K}\mathrm{\mathcal{M}}\hbox{-} $$ ) algorithm for expanding an algebraic number α = (α1, . . . , αd) from ℝd in a multidimensional continued fraction, i.e., in a sequence of rational numbers $$ \frac{P_a}{Q_a}=\left(\frac{P_1^a}{Q^a},\dots, \frac{P_d^a}{Q^a}\right),\kern1em a=1,2,3,\dots, $$ from ℚd with numerators $$ {P}_1^a,\dots, {P}_d^a\in \mathrm{\mathbb{Z}} $$ and a common denominator Qa = 1, 2, 3, . . . is proposed. The $$ \mathcal{K}\mathrm{\mathcal{M}} $$ -algorithm belongs to the class of tuned algorithms and is based on constructing localized Pisot units ζ > 1, for which the moduli of all the conjugates ζ(i) ≠ ζ are contained in the θ- neighborhood of the number ζ−1/d, where the parameter θ may take an arbitrary fixed value. It is proved that given a real algebraic point α of degree deg(α) = d + 1, the $$ \mathcal{K}\mathrm{\mathcal{M}} $$ -algorithm provides its approximation such that $$ \left|\alpha -\frac{P_a}{Q_a}\right|\le \frac{c}{Q_a^{1+\frac{1}{a}-\theta }} $$ for all a ≥ aα,θ, where the constants aα,θ > 0 and c = cα,θ > 0 are independent of a = 1, 2, 3, . . ., and the convergents $$ \frac{P_a}{Q_a} $$ are computed from a certain recurrence relation with constant coefficients, determined by the choice of the localized unit ζ. Bibliography: 19 titles. |
| Databáze: | OpenAIRE |
| Externí odkaz: |
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