On the distribution of points in projective space of bounded height

In this paper we consider the uniform distribution of points in compact met ric spaces. We assume that there exists a probability measure on the Borel subsets of the space which is invariant under a suitable group of isometrie s. In this setting we prove the analogue of Weyl's criterion and the Erdos- Turan an inequality by using orthogonal polynomials associated with the spa ce and the measure. In particular, we discuss the special case of projectiv e space over completions of number fields in some detail. An invariant meas ure in these projective spaces is introduced, and the explicit formulas for the orthogonal polynomials in this case are given. Finally, using the anal ogous Erdos-Turan inequality, we prove that the set of all projective point s over the number field with bounded Arakelov height is uniformly distribut ed with respect to the invariant measure as the bound increases.