The number of configurations in lattice point counting I

Huxley, Martin Neil and Žunić, Joviša 2010. The number of configurations in lattice point counting I. Forum Mathematicum 22 (1) , pp. 127-152. 10.1515/FORUM.2010.007

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Abstract

When a strictly convex plane set S moves by translation, the set J of points of the integer lattice that lie in S changes. The number K of equivalence classes of sets J under lattice translations (configurations) is bounded in terms of the area of the Brunn-Minkowski difference set of S. If S satisfies the Triangle Condition, that no translate of S has three distinct lattice points in the boundary, then K is asymptotically equal to the area of the difference set, with an error term like that in the corresponding lattice point problem. If S satisfies a Smoothness Condition but not the Triangle Condition, then we obtain a lower bound for K, but not of the right order of magnitude. The case when S is a circle was treated in our earlier paper by a more complicated method. The Triangle Condition was removed by considerations of norms of Gaussian integers, which are special to the circle.

Item Type:	Article
Date Type:	Publication
Status:	Published
Schools:	Mathematics
Subjects:	Q Science > QA Mathematics
Additional Information:	PDF uploaded nin accordance with publisher's policy as of 28/07/14.
Publisher:	de Gruyter
ISSN:	0933-7741
Date of First Compliant Deposit:	30 March 2016
Last Modified:	15 May 2023 20:56
URI:	https://orca.cardiff.ac.uk/id/eprint/13888

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