An Optimal, Stable Continued Fraction Algorithm

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) 2Rn this CFA generates a sequence of vectors (p (k) 1 : : : p (k) n;1 q (k)) 2Zn k = 1 2 : : : with increasing integers jq (k)j satisfying for i = 1 : : : n ; 1 jxi ; pi (k)=q (k)j 2(n+2)=4 p 1 + x 2 i = jq (k)j1+ 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.

• Type: Conference
• Year: 1996
• Where: IPCO
• Authors: Carsten Rössner, Claus-Peter Schnorr
• Keywords: Arbitrary Dimension, Fraction Algorithm, IPCO 1996, IPCO 2007, Simultaneous Diophantine Approximations