Linear-Fractional Invariance of the Simplex-Module Algorithm for Expanding Algebraic Numbers in Multidimensional Continued Fractions

Abstract

The paper establishes the invariance of the simplex-module algorithm for expanding real numbers α = (α₁, …, α_d) in multidimensional continued fractions under linear-fractional transformations \( {\alpha}^{\prime }=\left({\alpha}_1^{\prime },\dots, {\alpha}_d^1\right)=U\left\langle \alpha \right\rangle \) with matrices U from the unimodular group GL_d+1(ℤ). It is shown that the convergents of the transformed collections of numbers α^′ satisfy the same recurrence relation and have the same approximation order.