Well-rounded sublattices of planar lattices

Michael Baake ¹   ; Rudolf Scharlau ²   ; Peter Zeiner ¹

¹ Fakultät für Mathematik Universität Bielefeld Box 100131 33501 Bielefeld, Germany
² Fakultät für Mathematik Technische Universität Dortmund 44221 Dortmund, Germany

Acta Arithmetica, Tome 166 (2014) no. 4, pp. 301-334

Résumé

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.

Détail

DOI : 10.4064/aa166-4-1

Keywords: lattice euclidean d space called well rounded contains linearly independent vectors minimal length class lattices important various questions including sphere packing homology computations task enumerating well rounded sublattices given lattice interest already dimension has recently treated several authors paper analyse question closely spirit earlier work similar sublattices coincidence site sublattices combining explicit geometric considerations known techniques theory dirichlet series arrive after considerable amount computation asymptotic results number well rounded sublattices given index planar lattice symmetric lattices square hexagonal lattice present detailed results