The Computational Complexity of Simultaneous Diophantine Approximation Problems

Simultaneous Diophantine approximation in d dimensions deals with the approximation of a vector ${\bf \alpha } = (\alpha _1 , \cdots ,\alpha _d )$ of d real numbers by vectors of rational numbers all having the same denominator. This paper considers the computational complexity of algorithms to find good simultaneous approximations to a given vector $ {\bf \alpha} $ of d rational numbers. We measure the goodness of an approximation using the sup norm. We show that a result of H. W. Lenstra, Jr. produces polynomial-time algorithms to find sup norm best approximations to a given vector ${\bf \alpha} $ when the dimension d is fixed. We show that a recent algorithm of A. K. Lenstra, H. W. Lenstra, Jr. and L. Lovasz to find short vectors in an integral lattice can be used to find a good approximation to a given vector ${\bf \alpha} $ in d dimensions with a denominator $Q^ * $ satisfying $1 \leqq Q^ * \leqq 2^{d/2} N$ which is within a factor $\sqrt {5d} 2^{(d + 1)/2} $ of the best approximation with denominato...