The Best Approximation of Algebraic Numbers by Multidimensional Continued Fractions

The karyon-module ($\mathcal{K}\mathrm{\mathcal{M}}\hbox{-}$) algorithm for expanding an algebraic number α = (α₁, . . . , α_d) from ℝ^d 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 ℚ^d with numerators ${P}_1^a,\dots, {P}_d^a\in \mathrm{\mathbb{Z}}$ and a common denominator Q^a = 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 ≥ a_α,θ, where the constants a_α,θ > 0 and c = c_α,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.