Well-rounded sublattices of planar lattices

Michael Baake, Rudolf Scharlau, and Peter Zeiner. "Well-rounded sublattices of planar lattices." Acta Arithmetica 166.4 (2014): 301-334. <http://eudml.org/doc/279850>.

@article{MichaelBaake2014,
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.},
author = {Michael Baake, Rudolf Scharlau, Peter Zeiner},
journal = {Acta Arithmetica},
keywords = {geometry of numbers; planar lattices; sublattice enumeration; Dirichlet series generating functions; asymptotic growth},
language = {eng},
number = {4},
pages = {301-334},
title = {Well-rounded sublattices of planar lattices},
url = {http://eudml.org/doc/279850},
volume = {166},
year = {2014},
}

TY - JOUR
AU - Michael Baake
AU - Rudolf Scharlau
AU - Peter Zeiner
TI - Well-rounded sublattices of planar lattices
JO - Acta Arithmetica
PY - 2014
VL - 166
IS - 4
SP - 301
EP - 334
AB - 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.
LA - eng
KW - geometry of numbers; planar lattices; sublattice enumeration; Dirichlet series generating functions; asymptotic growth
UR - http://eudml.org/doc/279850
ER -