Well-rounded sublattices of planar lattices

• Autorzy: Michael Baake , Rudolf Scharlau , Peter Zeiner
• Języki publikacji: EN

Abstrakty

EN

A lattice in Euclidean d-space is called well-rounded if it contains d linearly independent vectors of minimal length. This class of lattices is important for various questions, including sphere packing or homology computations. The task of enumerating well-rounded sublattices of a given lattice is of interest already in dimension 2, and has recently been treated by several authors. In this paper, we analyse the question more closely in the spirit of earlier work on similar sublattices and coincidence site sublattices. Combining explicit geometric considerations with known techniques from the theory of Dirichlet series, we arrive, after a considerable amount of computation, at asymptotic results on the number of well-rounded sublattices up to a given index in any planar lattice. For the two most symmetric lattices, the square and the hexagonal lattice, we present detailed results.

Kategorie tematyczne

• 52C05: Lattices and convex bodies in 2 dimensions
• 11H06: Lattices and convex bodies
• 05A15: Exact enumeration problems, generating functions

• Wydawca: Institute of Mathematics Polish Academy of Sciences
• Czasopismo: Acta Arithmetica
• Rocznik: 2014
• Tom: 166
• Numer: 4
• Strony: 301-334

Daty

wydano

2014

Twórcy

autor

Michael Baake

• Fakultät für Mathematik, Universität Bielefeld, Box 100131, 33501 Bielefeld, Germany

autor

Rudolf Scharlau

• Fakultät für Mathematik, Technische Universität Dortmund, 44221 Dortmund, Germany

autor

Peter Zeiner

• Fakultät für Mathematik, Universität Bielefeld, Box 100131, 33501 Bielefeld, Germany

Identyfikatory

DOI

10.4064/aa166-4-1