Disjoint spheres, approximation by imaginary quadratic numbers, and the logarithm law for geodesics

Abstract

This paper is based on the principle that probabalistic independence of certain sets in Euclidean space is forced by a disjoint collection of spheres in a Euclidean space of one higher dimension. (See Figure This principle allows a new proof of (a new variant of) Khintchine's approximation theorem for almost all reals by rationals w 3. The new proof extends naturally to the approximation of almost all complex numbers by ratios of integers p/q,p, q E 0(~ / -d ) in imaginary quadratic fields.