A multidimensional continued fraction and some of its statistical properties

The problem of simultaneously approximating a vector of irrational numbers with rationals is analyzed in a geometrical setting using notions of dynamical systems theory. We discuss here a (vectorial) multidimensional continued-fraction algorithm (MCFA) of additive type, the generalized mediant algorithm (GMA), and give a geometrical interpretation to it. We calculate the invariant measure of the GMA shift as well as its Kolmogorov-Sinai (KS) entropy for arbitrary number of irrationals. The KS entropy is related to the growth rate of denominators of the Euclidean algorithm. This is the first analytical calculation of the growth rate of denominators for any MCFA.Glossary L⁺ set of positive integers - [.] Gauss integer symbol (Section 2) - h entropy - I of irrationals to be simultaneously approximated - d dimension of the vector of convergents (equal to I+1) - ^P unit hypercube inp dimensions - support of the invariant measure (see Section 5) - E_ij elementary matrix, with klth component_kl+_ikjl - E-string product of elementary matrices given by the algorithm - verticesV_i corners of the elementary simplex adjoined to the origin (Section 3) - mediantsM_ik a direct sum of any two of the vertices (Section 3) - focus sum of all the vertices (Section 3) - Euclidean reverse of the E-string procedure (see Section 2) algorithm - OCF ordinary continued-fraction algorithm - GMA generalized mediant algorithm: the subject of this paper - JP Jacobi-Perron: the most well-studied MCFA - MCFA Multidimensional continued-fraction algorithm - KS entropy Kolmogorov-Sinai entropy - T_OCF ordinary continued-fraction shift map - FS Farey shift map - (a,..., z) irrational vector withI components; each element is an irrational - d(x) invariant measure - (x) invariant density =d(x)/dx] - ₁, ₂,...,_d thed Oseledec eigenvalues of the E-string (see Section 4) ordered ₁>1>₂₃... - ₁,...,_d–1 Oseledec eigenvalues of the shift map (Section 4) ordered greatest to smallest; all the _i>1, and _i=₁/_di+1 - ln ₁,..., In _{d– 1} Oseledecexponents of the shift map (Section 4) - Perm a permutation matrix (Section 4)