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

DOI:
https://doi.org/10.1090/S0002-9947-99-02275-8

Published electronically:
September 9, 1999

MathSciNet review:
1491857

Full-text PDF

Abstract | References | Similar Articles | Additional Information

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.

**1.**E. Bombieri, A.J. van der Poorten and J. D. Vaaler.*Effective Measures of Irrationality for Cubic Extensions of Number Fields,*Ann Scuola Norm. Sup. Pisa Cl. Sci. (4)**23**(1996), 211-248. MR**98d:11083****2.**E. Bombieri and J. Vaaler.*On Siegel's Lemma,*Invent. Math.**73**, 11-32 (1983). MR**85g:11049a****3.**J.W.S. Cassels. Local Fields. (LMSST 3) Cambridge Univ. Press (1986). MR**87i:11172****4.**K.K. Choi.*Diophantine Approximation on Projective Spaces over Number Fields,*Ph. D. Dissertation. The University of Texas at Austin (1996).**5.**K.K. Choi and J. D. Vaaler.*Diophantine Approximation in Projective Space,*Submitted for publication.**6.**S. Lang. Fundamentals of Diophantine Geometry, Springer-Verlag, New York, 1983. MR**85j:11005****7.**P. Grabner.*Erdös-Turán Type Discrepancy Bounds*, Monatsh. Math.**111**, 127-135 (1991). MR**92f:11108****8.**L. Kuipers and H. Niederreiter. Uniform Distribution of Sequences. John Wiley & Sons (1974). MR**54:7415****9.**R.S. Rumely. Capacity Theory on Algebraic Curves. Lecture Notes in Mathematics, vol. 1378, Springer-Verlag, New York, 1989. MR**91b:14018****10.**S. Schanuel.*Heights in Number Fields,*Bull. Soc. Math. France**107**(1979), 433-449. MR**81c:12025****11.**G. Szegö. Orthogonal Polynomials. Colloquium Publications Vol. 23, Amer. Math. Soc. (1991). MR**51:8724****12.**J. Thunder.*An Asymptotic Estimate for Heights of Algebraic Subspaces,*Trans. Amer. Math. Soc.**331**, 395-424 (1992). MR**92g:11062****13.**J. Thunder.*The Number of Solutions of Bounded Height to a System of Linear Equations,*J. Number Theory**43**, 228-250 (1993). MR**94a:11045****14.**S. Tyler.*The Lagrange Spectrum in Projective Space over a Local Field,*Ph. D. Dissertation. The University of Texas at Austin (1994).**15.**J. D. Vaaler.*Some Extremal Functions in Fourier Analysis,*Bull. Amer. Math. Soc.**12**, 183-216 (1985). MR**86g:42005**

Retrieve articles in *Transactions of the American Mathematical Society*
with MSC (1991):
11J61,
11J71,
11K60

Retrieve articles in all journals with MSC (1991): 11J61, 11J71, 11K60

Additional Information

**Kwok-Kwong Choi**

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

Email:
choi@maths.hku.hk

DOI:
https://doi.org/10.1090/S0002-9947-99-02275-8

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