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• W2091330756 abstract "This paper presents for the first time a higher-dimensional continued fraction algorithm (abbreviated “cfa”) that produces diophantine approximations of more than linear goodness. On input $x_1 , cdots ,x_{n - 1} in {bf R}$, it produces vectors $(p_1^{(k)} , cdots p_{n - 1}^{(k)} ,q^{(k)} ) in {bf Z}^n ,k = 1,2, cdots $, such that [ maxlimits_{1 leq i leq n - 1} left| x_i - frac{p_i^{(k)}}{q^{(k)} } right| leq frac{||x|| cdot {text{const}}(n)}{| q^{(k)} |^{1 + 1/(2n(n - 1))}}. ] By a theorem of Dirichlet, there is no algorithm that replaces the term $1/(2n(n - 1))$ by a term bigger than $1/(n - 1)$. The higher-dimensional cfa’s analyzed so far do not achieve better than $max_{1 leq i leq n - 1} |x_i - p_i^{(k)} /q^{(k)} |leq o(1)/|q^{(k)} |$. The $o(1)$ term decreases with k but is not known to be related with $q^{(k)} $. Other properties of the cfa are also generalized by the algorithm. On input $x_1 , cdots ,x_{n - 1} $ it starts with the standard basis of ${bf Z}^n $ and then constructs by performing elementary basis transformations a sequence $(mathcal{B}^{(k)} )_{k} $ of bases of ${bf Z}^n $. The sequence $(mathcal{B}^{(k)} )_{k} $ is finite if and only if the numbers $x_1 , cdots ,x_{n - 1} $, 1 are ${bf Z}$-linearly dependent; a linear dependence is found in case of existence. The maximal distance between the vectors of $mathcal{B}^{(k)} $ and the straight line $(x_1 , cdots, x_{n - 1} ,1)$${bf R}$, tends to zero exponentially fast in k. For each k, the above-mentioned vector $(p_1^{(k)} , cdots ,p_{n - 1}^{(k)} ,q^{(k)} )$ is the first vector of basis $mathcal{B}^{(k)} $. The algorithm is a variant of an algorithm for the integer relation problem presented in [G. Bergman, Notes on Ferguson and Forcade’s Generalized Euclidean Algorithm, preprint, Univ. California, Berkeley, 1980] and analyzed in [J. Hastad, B. Just, J. Lagarias, and C. P Schnorr, SIAM J. Comput.,18 (1989), pp. 859–881]. The bound on the goodness of the diophantine approximations is proven with a “parallel induction” technique." @default.
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• W2091330756 date "1992-10-01" @default.
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• W2091330756 title "Generalizing the Continued Fraction Algorithm to Arbitrary Dimensions" @default.
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• W2091330756 doi "https://doi.org/10.1137/0221054" @default.
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