| 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: | 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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