ON THE DISTRIBUTION OF POINTS IN PROJECTIVE SPACE OF BOUNDED HEIGHT

In this paper we consider the uniform distribution of points in compact metric spaces. We assume that there exists a probability measure on the Borel subsets of the space which is invariant under a suitable group of isometries. In this setting we prove the analogue of Weyl's criterion and the Erdos-Tur an inequality by using orthogonal polynomials associated with the space and the measure. In particular, we discuss the special case of pro- jective space over completions of number elds in some detail. An invariant measure in these projective spaces is introduced, and the explicit formulas for the orthogonal polynomials in this case are given. Finally, using the analo- gous Erdos-Tur an inequality, we prove that the set of all projective points over the number eld with bounded Arakelov height is uniformly distributed with respect to the invariant measure as the bound increases.