Generalizing the Continued Fraction Algorithm to Arbitrary Dimensions

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 \[ \max\limits_{1 \leq i \leq n - 1} \left| x_i - \frac{p_i^{(k)}}{q^{(k)} } \right| \leq \frac{||x|| \cdot {\text{const}}(n)}{| q^{(k)} |^{1 + 1/(2n(n - 1))}}. \] 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.