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Well-rounded sublattices of planar lattices

Volume 166 / 2014

Michael Baake, Rudolf Scharlau, Peter Zeiner Acta Arithmetica 166 (2014), 301-334 MSC: Primary 11H06; Secondary 05A15, 52C05. DOI: 10.4064/aa166-4-1

Abstract

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.

Authors

  • Michael BaakeFakultät für Mathematik
    Universität Bielefeld
    Box 100131
    33501 Bielefeld, Germany
    e-mail
  • Rudolf ScharlauFakultät für Mathematik
    Technische Universität Dortmund
    44221 Dortmund, Germany
    e-mail
  • Peter ZeinerFakultät für Mathematik
    Universität Bielefeld
    Box 100131
    33501 Bielefeld, Germany
    e-mail

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