Diophantine approximations of rationals and related problems
by
Iskander Aliev
Institute of Discrete Mathematics and Geometry,TU Wien, Austria

In the present talk we discuss recent results on simultaneous Diophantine approximations of rationals by rationals with smaller denominators. Roughly speaking, given a rational point r = (p₁/q, ... , p_n/q) with gcd(p₁, ... , p_n, q)=1, we study the lattice L(r) of rational approximations to r with smaller denominators. Then for any rational lattice L, we construct a sequence of rational points r_t such that the sequence of corresponding lattices L(r_t) after an appropriate normalization tends to L. As an application, we give new estimates for optimal constants in the problem of decomposition of integer vectors proposed by A. Schinzel and in the problem of Diophantine approximations under a constraint on the denominator proposed by W. B. Jurkat.

Date received: April 28, 2004