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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
Published electronically: June 9, 2011
MathSciNet review: 2846337
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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

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.

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