Publications Meetings The Profession Membership Programs Math Samplings Policy & Advocacy In the News About the AMS
   
Mobile Device Pairing
Green Open Access
Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

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
Full-text PDF

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.


References [Enhancements On Off] (What's this?)


Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC (2010): 52A30, 11H06, 11P21, 11D45

Retrieve articles in all journals with MSC (2010): 52A30, 11H06, 11P21, 11D45


Additional Information

Martin Widmer
Affiliation: Department of Mathematics, Technische Universität Graz, Steyrergasse 30/II, 8010 Graz, Austria
Email: widmer@math.tugraz.at

DOI: http://dx.doi.org/10.1090/S0002-9939-2011-10926-2
PII: S 0002-9939(2011)10926-2
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.