Lipschitz class, narrow class, and counting lattice points
Author:
Martin Widmer
Journal:
Proc. Amer. Math. Soc. 140 (2012), 677-689
MSC (2010):
Primary 52A30, 11H06; Secondary 11P21, 11D45
DOI:
https://doi.org/10.1090/S0002-9939-2011-10926-2
Published electronically:
June 9, 2011
MathSciNet review:
2846337
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Abstract | References | Similar Articles | Additional Information
Abstract: A well-known principle says that the number of lattice points in a bounded subset $S$ of Euclidean space is about the ratio of the volume and the lattice determinant, subject to some relatively mild conditions on $S$. In the literature one finds two different types of such conditions: one asserts the Lipschitz parameterizability of the boundary $\partial S$, and the other one is based on intersection properties of lines with $S$ and its projections to linear subspaces. We compare these conditions and address a question, which we answer in some special cases. Then we give some simple upper bounds on the number of lattice points in a convex set, and finally, we apply these results to obtain estimates for the number of rational points of bounded height on certain projective varieties.
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Additional Information
Martin Widmer
Affiliation:
Department of Mathematics, Technische Universität Graz, Steyrergasse 30/II, 8010 Graz, Austria
Email:
widmer@math.tugraz.at
Keywords:
Lattice points,
counting,
Lipschitz class,
narrow class,
height
Received by editor(s):
August 3, 2010
Received by editor(s) in revised form:
November 23, 2010
Published electronically:
June 9, 2011
Communicated by:
Matthew A. Papanikolas
Article copyright:
© Copyright 2011
American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.
