Well-Rounded Sublattices and Coincidence Site Lattices

Zeiner, Peter

doi:10.1007/978-94-007-6431-6_6

Cited by 2 publications

(30 citation statements)

References 11 publications

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“…In our terminology, commensurateness means that Γ 1 ∩ Γ 2 is a sublattice (of full rank) of both Γ 1 and Γ 2 . Actually, there are several ways to characterise commensurateness [98]. Lemma 2.2.…”

Section: Preliminaries On Latticesmentioning

confidence: 99%

“…This implies that M 1 and M 2 can only be commensurate if they have the same rank. Once we know that two embedded modules in R d have the same rank, the situation becomes easier as we can characterise commensurateness in several ways [98], which we recall here. Lemma 6.4.…”

Section: Similar Submodulesmentioning

confidence: 99%

“…Lemma 7.9 provides us with some useful bounds on the coincidence indices of a sublattice. In certain cases, we can even get sharper bounds [65,98]. As an example, we mention the following result, which is a special case of [98, Thm.…”

Section: Similar Submodulesmentioning

confidence: 99%

“…In fact, intersections of more than two isometric commensurate copies of a lattice have already been discussed in [8,95,18,98]. Let us first recall the corresponding definitions.…”

Section: Similar Submodulesmentioning

confidence: 99%

“…In general, however, the counting functions fail to be multiplicative, though we have the following weaker result. Theorem 7.17 ([96,98]). The arithmetic function c iso Γ (n), c rot Γ (n) and c Γ (n) are supermultiplicative, that is, c iso Γ (mn) ≥ c iso Γ (m) c iso Γ (n) holds for coprime integers m and n, and likewise for the other functions.…”

Section: Similar Submodulesmentioning

confidence: 99%

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Geometric Enumeration Problems for Lattices and Embedded ℤ-Modules

Baake¹,

Zeiner²

Aperiodic Order

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In this review, we count and classify certain sublattices of a given lattice, as motivated by crystallography. We use methods from algebra and algebraic number theory to find and enumerate the sublattices according to their index. In addition, we use tools from analytic number theory to determine the asymptotic behaviour of the corresponding counting functions. Our main focus lies on similar sublattices and coincidence site lattices, the latter playing an important role in crystallography. As many results are algebraic in nature, we also generalise them to Z-modules embedded in R d . R d , which led to the notion of coincidence site modules (CSMs). They are used to describe grain boundaries in quasicrystals; compare [15,72,88] and references therein.This new development also triggered a more detailed study of lattices in dimensions d > 3, as they are used to generate aperiodic point sets by the now common cut and project technique; compare [9, Ch. 7]. In particular, lattices in dimension d = 4 such as the hypercubic lattices [4,94] and the A 4 -lattice [11,56] were studied.Further applications of CSLs can be found in coding theory in connection with so-called lattice quantisers, where lattices in large dimensions and with high packing densities are important; compare [34,85] for general background, as well as [1] for concrete applications of the A 4 -lattice and [2] for the hexagonal lattice. However, not much is known about lattices in dimensions d > 5, although there are some partial results for rational lattices [99,100,57].The original concept of CSLs has been generalised in several ways. In particular, one may study the intersection of several rotated copies of a lattice, which are known as multiple CSLs; compare [8,95,18]. They have applications to so-called multiple junctions [40,41,42], which are multiple crystal grains meeting at some common manifold. Whereas classical CSLs involve only linear isometries, one may consider affine isometries as well, which is directly related to the question of coincidences of crystallographic point packings; compare [64,66,63]. The latter are connected to the problem of coincidences of coloured lattices and colour coincidences [65,63,67].The planar case is certainly the best studied. Here, also a connection between CSLs and well-rounded sublattices has been established [17]. Moreover, even some results for the hyperbolic plane [78] have been found.Naturally, CSLs are not the only sublattices that are of interest in crystallography and coding theory. Classifying sublattices with certain symmetry constraints has a long tradition in mathematics and in crystallography; compare [79,80] and references therein. An interesting question is the number of sublattices that are similar to its parent lattice. It has been answered in detail for a considerable collection of lattices [7,14,12] in dimensions d ≤ 4. For higher dimensions, some existence results have been obtained by Conway, Rains and Sloane, who were motivated by problems in coding theory [26].Actually, some years ago, a cl...

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Section: Preliminaries On Latticesmentioning

confidence: 99%

Section: Similar Submodulesmentioning

confidence: 99%

Section: Similar Submodulesmentioning

confidence: 99%

“…In fact, intersections of more than two isometric commensurate copies of a lattice have already been discussed in [8,95,18,98]. Let us first recall the corresponding definitions.…”

Section: Similar Submodulesmentioning

confidence: 99%

Section: Similar Submodulesmentioning

confidence: 99%

See 3 more Smart Citations

Geometric Enumeration Problems for Lattices and Embedded ℤ-Modules

Baake¹,

Zeiner²

Aperiodic Order

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Well-rounded sublattices of planar lattices

2014

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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 wellrounded 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.

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Well-Rounded Sublattices and Coincidence Site Lattices

Cited by 2 publications

References 11 publications

Geometric Enumeration Problems for Lattices and Embedded ℤ-Modules

Geometric Enumeration Problems for Lattices and Embedded ℤ-Modules

Well-rounded sublattices of planar lattices

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