The best approximation of algebraic numbers by multidimensional continued fractions

V. G. Zhuravlev

Zapiski Nauchnykh Seminarov POMI, Algebra and number theory. Part 2, Tome 479 (2019), pp. 52-84

Résumé

A karyon-modular algorithm ($\mathcal {KM}$-algorithm) is proposed for decomposition of algebraic numbers $\alpha = (\alpha_1, \ldots, \alpha_d)$ from $\mathbb {R}^{d}$ to multidimensional continued fractions, that are a sequence of rational numbers $$ \frac{P_{a}}{Q_{a}}=\Bigl( \frac{P^{a}_1}{Q^{a}},\ldots,\frac{P^{a}_d}{Q^{a}}\Bigr), a=1,2,3,\ldots, $$ from $\mathbb{Q}^d$ with numerators $P^{a}_1,\ldots,P^{a}_d \in \mathbb{Z}$ and the common denominator $Q^{a}=1,2,3,\ldots$ The $ \mathcal{KM}$-algorithm belongs to a class of tuning algorithms. It is based on the construction of localized Pisot units $\zeta>1$, for which the moduli of all conjugates $\zeta^{(i)}\ne \zeta $ are contained in the $ \theta $-neighbourhood of the number $ \zeta^{- 1 /d}$, where the parameter $ \theta> 0 $ can take an arbitrary fixed value. It is proved that if $ \alpha $ is a real algebraic point of degree $ \mathrm {deg} (\alpha) = d + 1 $, then  $ \mathcal {KM} $ - algorithm allows to obtain the following approximation $$ \Bigl | \alpha - \frac {P_{a}}{Q_{a}} \Bigr | \leq \frac {c} {Q^{1+ \frac{1}{d} - \theta}_{a}} $$ for all $ a \geq a_{\alpha, \theta} $, where the constants $ a_{\alpha, \theta}> 0 $ and $ c = c_{\alpha, \theta}> 0 $ do not depend on $ a = 1,2,3, \ldots $ and the convergent fractions $ \frac {P_{a}} {Q_{a}} $ are calculated by means of some recurrence relation with constant coefficients determined by the choice of the localized units $ \zeta $.