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
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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© 1996 Springer-Verlag Berlin Heidelberg
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Rössner, C., Schnorr, C.P. (1996). An optimal, stable continued fraction algorithm for arbitrary dimension. In: Cunningham, W.H., McCormick, S.T., Queyranne, M. (eds) Integer Programming and Combinatorial Optimization. IPCO 1996. Lecture Notes in Computer Science, vol 1084. Springer, Berlin, Heidelberg. https://doi.org/10.1007/3-540-61310-2_3
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DOI: https://doi.org/10.1007/3-540-61310-2_3
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