An Optimal, Stable Continued Fraction Algorithm for Arbitrary Dimension

We analyse a continued fraction algorithm (abbreviated CFA) for arbitrary dimension n showing that it produces simultaneous diophantine approximations which are up to the factor 2 (n+2)=4 best possible. Given a real vector x =(x1; : : : ; xn?1 ; 1) 2R n this CFA generates a sequence of vectors (p (k) jxi ? pi (k) =q (k) j 2 (n+2)=4 p 1 + x 2 i = jq (k) j 1+ 1 n?1 : By a theorem of Dirichlet this bound is best possible in that the exponent 1 + 1 n?1 can in general not be increased.