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An optimal, stable continued fraction algorithm for arbitrary dimension

  • Carsten Rössner
  • Claus P. Schnorr
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 1084)

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 =(x1,..., xn−1, 1) εℝn this CFA generates a sequence of vectors (p1(k))..., pn−1(k), qk) 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.

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Copyright information

© Springer-Verlag Berlin Heidelberg 1996

Authors and Affiliations

  • Carsten Rössner
    • 1
  • Claus P. Schnorr
    • 1
  1. 1.Fachbereich Mathematik/InformatikUniversität FrankfurtFrankfurt am MainGermany

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