Linear-Fractional Invariance of Multidimensional Continued Fractions

Abstract

The invariance of the simplex-karyon algorithm for expanding real numbers α = (α₁, …, α_d) in multidimensional continued fractions under linear-fractional transformations \( {\alpha}^{\prime }=\left({\alpha}_1^{\prime },\dots, {\alpha}_d^{\prime}\right)=U\left\langle \alpha \right\rangle \) with matrices U from the unimodular group GL_d+1(ℤ) is established. For the transformed collections α^′, convergents of the best approximations are found.