On the distribution of points in projective space of bounded height
Author:
KwokKwong Choi
Journal:
Trans. Amer. Math. Soc. 352 (2000), 10711111
MSC (1991):
Primary 11J61, 11J71, 11K60
Published electronically:
September 9, 1999
MathSciNet review:
1491857
Fulltext PDF Free Access
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 ErdösTurán 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 ErdösTurán 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), no. 2, 211–248. MR 1433423
(98d:11083)
 2.
E.
Bombieri and J.
Vaaler, On Siegel’s lemma, Invent. Math.
73 (1983), no. 1, 11–32. MR 707346
(85g:11049a), http://dx.doi.org/10.1007/BF01393823
 3.
J.
W. S. Cassels, Local fields, London Mathematical Society
Student Texts, vol. 3, Cambridge University Press, Cambridge, 1986. MR 861410
(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.
Serge
Lang, Fundamentals of Diophantine geometry, SpringerVerlag,
New York, 1983. MR 715605
(85j:11005)
 7.
Peter
J. Grabner, ErdősTurán type discrepancy bounds,
Monatsh. Math. 111 (1991), no. 2, 127–135. MR 1100852
(92f:11108), http://dx.doi.org/10.1007/BF01332351
 8.
L.
Kuipers and H.
Niederreiter, Uniform distribution of sequences,
WileyInterscience [John Wiley & Sons], New YorkLondonSydney, 1974.
Pure and Applied Mathematics. MR 0419394
(54 #7415)
 9.
Robert
S. Rumely, Capacity theory on algebraic curves, Lecture Notes
in Mathematics, vol. 1378, SpringerVerlag, Berlin, 1989. MR 1009368
(91b:14018)
 10.
Stephen
Hoel Schanuel, Heights in number fields, Bull. Soc. Math.
France 107 (1979), no. 4, 433–449 (English,
with French summary). MR 557080
(81c:12025)
 11.
Gábor
Szegő, Orthogonal polynomials, 4th ed., American
Mathematical Society, Providence, R.I., 1975. American Mathematical
Society, Colloquium Publications, Vol. XXIII. MR 0372517
(51 #8724)
 12.
Jeffrey
Lin Thunder, An asymptotic estimate for heights of
algebraic subspaces, Trans. Amer. Math.
Soc. 331 (1992), no. 1, 395–424. MR 1072102
(92g:11062), http://dx.doi.org/10.1090/S00029947199210721020
 13.
Jeffrey
Lin Thunder, The number of solutions of bounded height to a system
of linear equations, J. Number Theory 43 (1993),
no. 2, 228–250. MR 1207503
(94a:11045), http://dx.doi.org/10.1006/jnth.1993.1021
 14.
S. Tyler. The Lagrange Spectrum in Projective Space over a Local Field, Ph. D. Dissertation. The University of Texas at Austin (1994).
 15.
Jeffrey
D. Vaaler, Some extremal functions in Fourier
analysis, Bull. Amer. Math. Soc. (N.S.)
12 (1985), no. 2,
183–216. MR
776471 (86g:42005), http://dx.doi.org/10.1090/S027309791985153492
 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), 211248. MR 98d:11083
 2.
 E. Bombieri and J. Vaaler. On Siegel's Lemma, Invent. Math. 73, 1132 (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, SpringerVerlag, New York, 1983. MR 85j:11005
 7.
 P. Grabner. ErdösTurán Type Discrepancy Bounds, Monatsh. Math. 111, 127135 (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, SpringerVerlag, New York, 1989. MR 91b:14018
 10.
 S. Schanuel. Heights in Number Fields, Bull. Soc. Math. France 107 (1979), 433449. 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, 395424 (1992). MR 92g:11062
 13.
 J. Thunder. The Number of Solutions of Bounded Height to a System of Linear Equations, J. Number Theory 43, 228250 (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, 183216 (1985). MR 86g:42005
Similar Articles
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
KwokKwong 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:
http://dx.doi.org/10.1090/S0002994799022758
PII:
S 00029947(99)022758
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
