On the distribution of points in projective space of bounded height
Author: Kwok-Kwong Choi
Journal: Trans. Amer. Math. Soc. 352 (2000), 1071-1111
MSC (1991): Primary 11J61, 11J71, 11K60
Published electronically: September 9, 1999
MathSciNet review: 1491857
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Abstract: 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 Erdös-Turán inequality by using orthogonal polynomials associated with the space and the measure. In particular, we discuss the special case of projective space over completions of number fields 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 analogous Erdös-Turán inequality, we prove that the set of all projective points over the number field with bounded Arakelov height is uniformly distributed with respect to the invariant measure as the bound increases.
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Affiliation: Department of Mathematics, Statistics, Simon Fraser University, Burnaby, British Columbia, V5A 1S6, Canada; Department of Mathematics, University of British Columbia, Vancouver, British Columbia, V6T 1Z2, Canada
Address at time of publication: Department of Mathematics, The University of Hong Kong, Pokfulam Road, Hong Kong
Received by editor(s): April 24, 1997
Received by editor(s) in revised form: December 18, 1997
Published electronically: September 9, 1999
Additional Notes: The author was supported by NSF Grant DMS 9304580
Article copyright: © Copyright 1999 American Mathematical Society