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Forum Mathematicum

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The number of configurations in lattice point counting I

Martin N. Huxley1 / Joviša Žunić2

1School of Mathematics, Cardiff University, 23 Senghennydd Road, Cardiff CF24 4AG, U.K.

2Department of Computer Science, University of Exeter, Harrison Building, Exeter EX4 4QF, U.K.

Citation Information: Forum Mathematicum. Volume 22, Issue 1, Pages 127–152, ISSN (Online) 1435-5337, ISSN (Print) 0933-7741, DOI: https://doi.org/10.1515/forum.2010.007, October 2009

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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.

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