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Proceedings of the American Mathematical Society

Published by the American Mathematical Society since 1950, Proceedings of the American Mathematical Society is devoted to shorter research articles in all areas of pure and applied mathematics.

ISSN 1088-6826 (online) ISSN 0002-9939 (print)

The 2020 MCQ for Proceedings of the American Mathematical Society is 0.85.

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Lipschitz class, narrow class, and counting lattice points
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by Martin Widmer PDF
Proc. Amer. Math. Soc. 140 (2012), 677-689 Request permission


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:
  • 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
  • © Copyright 2011 American Mathematical Society
    The copyright for this article reverts to public domain 28 years after publication.
  • Journal: Proc. Amer. Math. Soc. 140 (2012), 677-689
  • MSC (2010): Primary 52A30, 11H06; Secondary 11P21, 11D45
  • DOI:
  • MathSciNet review: 2846337