arXiv subjects

Peter Zeiner

Publications and source records attributed to Peter Zeiner.

15 recordsLinked to original sources

arXiv2018

The coincidence problem for shifted lattices and crystallographic point packings

A coincidence site lattice is a sublattice formed by the intersection of a lattice $Γ$ in $\mathbb{R}^d$ with the image of $Γ$ under a linear isometry. Such a linear isometry is referred to as a linear coincidence isometry of $Γ$. Here, we consider the more general case allowing any affine isometry. Consequently, general results on coincidence isometries of shifted copies of lattices, and of crystallographic point packings are obtained. In particular, we discuss the shifted square lattice and the diamond packing in detail.

math.MG↗

arXiv2017

Geometric enumeration problems for lattices and embedded $\mathbb{Z}$-modules

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 $\mathbb{Z}$-modules embedded in $\mathbb{R}^d$.

math.MG↗

arXiv2015

Coincidence indices of sublattices and coincidences of colorings

Even though a lattice and its sublattices have the same group of coincidence isometries, the coincidence index of a coincidence isometry with respect to a lattice $Λ_1$ and to a sublattice $Λ_2$ may differ. Here, we examine the coloring of $Λ_1$ induced by $Λ_2$ to identify how the coincidence indices with respect to $Λ_1$ and to $Λ_2$ are related. This leads to a generalization of the notion of color symmetries of lattices to what we call color coincidences of lattices. Examples involving the cubic and hypercubic lattices are given to illustrate these ideas.

math.MG↗

arXiv2014

Well-rounded sublattices of planar lattices

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.

math.NT↗

arXiv2014

Similar submodules and coincidence site modules

We consider connections between similar sublattices and coincidence site lattices (CSLs), and more generally between similar submodules and coincidence site modules of general (free) $\mathbb{Z}$-modules in $\mathbb{R}^d$. In particular, we generalise results obtained by S. Glied and M. Baake [1,2] on similarity and coincidence isometries of lattices and certain lattice-like modules called $\mathcal{S}$-modules. An important result is that the factor group $\mathrm{OS}(M)/\mathrm{OC}(M)$ is Abelian for arbitrary $\mathbb{Z}$-modules $M$, where $\mathrm{OS}(M)$ and $\mathrm{OC}(M)$ are the groups of similar and coincidence isometries, respectively. In addition, we derive various relations between the indices of CSLs and their corresponding similar sublattices. [1] S. Glied, M. Baake, Similarity versus coincidence rotations of lattices, Z. Krist. 223, 770--772 (2008). DOI: 10.1524/zkri.2008.1054 [2] S. Glied, Similarity and coincidence isometries for modules, Can. Math. Bull. 55, 98--107 (2011). DOI: 10.4153/CMB-2011-076-x

math.NT↗

arXiv2013

CSLs of the root lattice $\mathbf{A_4}$

Recently, the group of coincidence isometries of the root lattice $A_4$ has been determined providing a classification of these isometries with respect to their coincidence indices. A more difficult task is the classification of all CSLs, since different coincidence isometries may generate the same CSL. In contrast to the typical examples in dimensions $d \leq 3$, where coincidence isometries generating the same CSL can only differ by a symmetry operation, the situation is more involved in 4 dimensions. Here, we find coincidence isometries that are not related by a symmetry operation but nevertheless give rise to the same CSL. We indicate how the classification of CSLs can be obtained by making use of the icosian ring and provide explicit criteria for two isometries to generate the same CSL. Moreover, we determine the number of CSLs of a given index and encapsulate the result in a Dirichlet series generating function.

math.MG↗

arXiv2012

Multiplicativity in the theory of coincidence site lattices

Coincidence Site Lattices (CSLs) are a well established tool in the theory of grain boundaries. For several lattices up to dimension $d=4$, the CSLs are known explicitly as well as their indices and multiplicity functions. Many of them share a particular property: their multiplicity functions are multiplicative. We show how multiplicativity is connected to certain decompositions of CSLs and the corresponding coincidence rotations and present some criteria for multiplicativity. In general, however, multiplicativity is violated, while supermultiplicativity still holds.

math.MG↗

arXiv2012

Well-rounded sublattices and coincidence site lattices

A lattice is called well-rounded, if its lattice vectors of minimal length span the ambient space. We show that there are interesting connections between the existence of well-rounded sublattices and coincidence site lattices (CSLs). Furthermore, we count the number of well-rounded sublattices for several planar lattices and give their asymptotic behaviour.

math.MG↗

arXiv2010

Colourings of lattices and coincidence site lattices

The relationship between the coincidence indices of a lattice $Γ_1$ and a sublattice $Γ_2$ of $Γ_1$ is examined via the colouring of $Γ_1$ that is obtained by assigning a unique colour to each coset of $Γ_2$. In addition, the idea of colour symmetry, originally defined for symmetries of lattices, is extended to coincidence isometries of lattices. An example involving the Ammann-Beenker tiling is provided to illustrate the results in the quasicrystal setting.

math.MG↗

arXiv2010

Coincidence isometries of a shifted square lattice

We consider the coincidence problem for the square lattice that is translated by an arbitrary vector. General results are obtained about the set of coincidence isometries and the coincidence site lattices of a shifted square lattice by identifying the square lattice with the ring of Gaussian integers. To illustrate them, we calculate the set of coincidence isometries, as well as generating functions for the number of coincidence site lattices and coincidence isometries, for specific examples.

math.MG↗

arXiv2009

Similar sublattices of planar lattices

The similar sublattices of a planar lattice can be classified via its multiplier ring. The latter is the ring of rational integers in the generic case, and an order in an imaginary quadratic field otherwise. Several classes of examples are discussed, with special emphasis on concrete results. In particular, we derive Dirichlet series generating functions for the number of distinct similar sublattices of a given index, and relate them to various zeta functions of orders in imaginary quadratic fields.

math.MG↗

arXiv2007

Coincidence rotations of the root lattice $A_4$

The coincidence site lattices of the root lattice $A_4$ are considered, and the statistics of the corresponding coincidence rotations according to their indices is expressed in terms of a Dirichlet series generating function. This is possible via an embedding of $A_4$ into the icosian ring with its rich arithmetic structure, which recently (arXiv:math.MG/0702448) led to the classification of the similar sublattices of $A_4$.

math.MG↗

arXiv2006

Symmetries of Coincidence Site Lattices of Cubic Lattices

We consider the symmetries of coincidence site lattices of 3-dimensional cubic lattices. This includes the discussion of the symmetry groups and the Bravais classes of the CSLs. We derive various criteria and necessary conditions for symmetry operations of CSLs. They are used to obtain a complete list of the symmetry groups and the Bravais classes of those CSLs that are generated by a rotation through the angle $π$.

math.MG↗

arXiv2006

Coincidences of Hypercubic Lattices in 4 dimensions

We consider the CSLs of 4-dimensional hypercubic lattices. In particular, we derive the coincidence index $Σ$ and calculate the number of different CSLs as well as the number of inequivalent CSLs for a given $Σ$. The hypercubic face centered case is dealt with in detail and it is sketched how to derive the corresponding results for the primitive hypercubic lattice.

math.MG↗

arXiv2006

Multiple CSLs for the body centered cubic lattice

Ordinary Coincidence Site Lattices (CSLs) are defined as the intersection of a lattice $Γ$ with a rotated copy $RΓ$ of itself. They are useful for classifying grain boundaries and have been studied extensively since the mid sixties. Recently the interests turned to so-called multiple CSLs, i.e. intersections of $n$ rotated copies of a given lattice $Γ$, in particular in connection with lattice quantizers. Here we consider multiple CSLs for the 3-dimensional body centered cubic lattice. We discuss the spectrum of coincidence indices and their multiplicity, in particular we show that the latter is a multiplicative function and give an explicit expression of it for some special cases.

math.MG↗