The number of configurations in lattice point counting I

✍ Scribed by Huxley, Martin N.; Žunić, Joviša

• Book ID: 126674650
• Publisher: Walter de Gruyter GmbH & Co. KG
• Year: 2010
• Tongue: English
• Weight: 310 KB
• Volume: 22
• Category: Article
• ISSN: 0933-7741
• DOI: 10.1515/forum.2010.007

✦ Synopsis

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.