229 research outputs found
Similar Sublattices and Coincidence Rotations of the Root Lattice A4 and its Dual
A natural way to describe the Penrose tiling employs the projection method on
the basis of the root lattice A4 or its dual. Properties of these lattices are
thus related to properties of the Penrose tiling. Moreover, the root lattice A4
appears in various other contexts such as sphere packings, efficient coding
schemes and lattice quantizers.
Here, the lattice A4 is considered within the icosian ring, whose rich
arithmetic structure leads to parametrisations of the similar sublattices and
the coincidence rotations of A4 and its dual lattice. These parametrisations,
both in terms of a single icosian, imply an index formula for the corresponding
sublattices. The results are encapsulated in Dirichlet series generating
functions. For every index, they provide the number of distinct similar
sublattices as well as the number of coincidence rotations of A4 and its dual.Comment: 8 pages, paper presented at ICQ10 (Zurich, Switzerland
Multi-Component Model Sets and Invariant Densities
Model sets (also called cut and project sets) are generalizations of
lattices, and multi-component model sets are generalizations of lattices with
colourings. In this paper, we study self-similarities of multi-component model
sets. The main point may be simply summarized: whenever there is a
self-similarity, there are also naturally related density functions. As in the
case of ordinary model sets, we show that invariant densities exist and that
they produce absolutely continuous invariant measures in internal space, these
features now appearing in matrix form. We establish a close connection between
the theory of invariant densities and the spectral theory of matrix continuous
refinement operators.Comment: 12 pages, 2 figures, to appear in: Aperiodic 9
Self-Similarities and Invariant Densities for Model Sets
Model sets (also called cut and project sets) are generalizations of
lattices. Here we show how the self-similarities of model sets are a natural
replacement for the group of translations of a lattice. This leads us to the
concept of averaging operators and invariant densities on model sets. We prove
that invariant densities exist and that they produce absolutely continuous
invariant measures in internal space. We study the invariant densities and
their relationships to diffraction, continuous refinement operators, and
Hutchinson measures.Comment: 15 pages, 2 figures, to appear in: Algebraic Methods and Theoretical
Physics (ed. Y. St. Aubin
Invariant Submodules and Semigroups of Self-Similarities for Fibonacci Modules
The problem of invariance and self-similarity in Z-modules is investigated.
For a selection of examples relevant to quasicrystals, especially Fibonacci
modules, we determine the semigroup of self-similarities and encapsulate the
number of similarity submodules in terms of Dirichlet series generating
functions.Comment: 7 pages; to appear in: Aperiodic 97, eds. M. de Boissieu, J. L.
Verger-Gaugry and R. Currat, World Scientific, Singapore (1998), in pres
Characterizations of model sets by dynamical systems
It is shown how regular model sets can be characterized in terms of
regularity properties of their associated dynamical systems. The proof proceeds
in two steps. First, we characterize regular model sets in terms of a certain
map and then relate the properties of to ones of the underlying
dynamical system. As a by-product, we can show that regular model sets are, in
a suitable sense, as close to periodic sets as possible among repetitive
aperiodic sets.Comment: 41 pages, revised versio
Similar sublattices of the root lattice
Similar sublattices of the root lattice are possible, according to a
result of Conway, Rains and Sloane, for each index that is the square of a
non-zero integer of the form . Here, we add a constructive
approach, based on the arithmetic of the quaternion algebra and the existence of a particular involution of the
second kind, which also provides the actual sublattices and the number of
different solutions for a given index. The corresponding Dirichlet series
generating function is closely related to the zeta function of the icosian
ring.Comment: 17 pages, 1 figure; revised version with several additions and
improvement
Weighted Dirac combs with pure point diffraction
A class of translation bounded complex measures, which have the form of
weighted Dirac combs, on locally compact Abelian groups is investigated. Given
such a Dirac comb, we are interested in its diffraction spectrum which emerges
as the Fourier transform of the autocorrelation measure. We present a
sufficient set of conditions to ensure that the diffraction measure is a pure
point measure. Simultaneously, we establish a natural link to the theory of the
cut and project formalism and to the theory of almost periodic measures. Our
conditions are general enough to cover the known theory of model sets, but also
to include examples such as the visible lattice points.Comment: 44 pages; several corrections and improvement
Diffraction from visible lattice points and k-th power free integers
We prove that the set of visible points of any lattice of dimension at least
2 has pure point diffraction spectrum, and we determine the diffraction
spectrum explicitly. This settles previous speculation on the exact nature of
the diffraction in this situation, see math-ph/9903046 and references therein.
Using similar methods we show the same result for the 1-dimensional set of k-th
power free integers with k at least 2. Of special interest is the fact that
neither of these sets is a Delone set --- each has holes of unbounded inradius.
We provide a careful formulation of the mathematical ideas underlying the study
of diffraction from infinite point sets.Comment: 45 pages, with minor corrections and improvements; dedicated to
Ludwig Danzer on the occasion of his 70th birthda
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