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)
| xi - pi (k) /q(k) | \leqslant 2(n + 2)/4 Ö{1 + xi2 } /| q(k) |1 + \tfrac1n - 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 + \tfrac1n - 11 + \tfrac{1}{{n - 1}}
can in general not be increased.