Lipschitz class, narrow class, and counting lattice points

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Lipschitz class, narrow class, and counting lattice points

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by Martin Widmer

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BibTeX

@MISC{Widmer_lipschitzclass,,
    author = {Martin Widmer},
    title = {Lipschitz class, narrow class, and counting lattice points},
    year = {}
}

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Abstract

Abstract. A well known principle says that the number of lattice points in a bounded subsets 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 ∂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. 1.

Keyphrases

lattice point lipschitz class narrow class bounded height certain projective variety different type euclidean space intersection property lattice determinant lipschitz parameterizability simple upper bound bounded subset special case rational point mild condition

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