An optimal, stable continued fraction algorithm for arbitrary dimension

Abstract

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 =(x ₁,..., x _n−1, 1) εℝⁿ this CFA generates a sequence of vectors (p ₁ ^(k))..., p _n−1 ^(k), q ^k) εℤⁿ, k = 1, 2,... with increasing integers ¦q ^(k)¦ satisfying for i = 1,..., n − 1

$$\left| {x_i - p_i ^{(k)} /q^{(k)} } \right| \leqslant 2^{(n + 2)/4} \sqrt {1 + x_i^2 } /\left| {q^{(k)} } \right|^{1 + \tfrac{1}{{n - 1}}} .$$

By a theorem of Dirichlet this bound is best possible in that the exponent $1 + \tfrac{1}{{n - 1}}$can in general not be increased.