Abstract
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.
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