325 research outputs found
Central sets and substitutive dynamical systems
In this paper we establish a new connection between central sets and the
strong coincidence conjecture for fixed points of irreducible primitive
substitutions of Pisot type. Central sets, first introduced by Furstenberg
using notions from topological dynamics, constitute a special class of subsets
of \nats possessing strong combinatorial properties: Each central set
contains arbitrarily long arithmetic progressions, and solutions to all
partition regular systems of homogeneous linear equations. We give an
equivalent reformulation of the strong coincidence condition in terms of
central sets and minimal idempotent ultrafilters in the Stone-\v{C}ech
compactification \beta \nats . This provides a new arithmetical approach to
an outstanding conjecture in tiling theory, the Pisot substitution conjecture.
The results in this paper rely on interactions between different areas of
mathematics, some of which had not previously been directly linked: They
include the general theory of combinatorics on words, abstract numeration
systems, tilings, topological dynamics and the algebraic/topological properties
of Stone-\v{C}ech compactification of \nats.Comment: arXiv admin note: substantial text overlap with arXiv:1110.4225,
arXiv:1301.511
Few smooth d-polytopes with n lattice points
We prove that, for fixed n there exist only finitely many embeddings of
Q-factorial toric varieties X into P^n that are induced by a complete linear
system. The proof is based on a combinatorial result that for fixed nonnegative
integers d and n, there are only finitely many smooth d-polytopes with n
lattice points. We also enumerate all smooth 3-polytopes with at most 12
lattice points. In fact, it is sufficient to bound the singularities and the
number of lattice points on edges to prove finiteness.Comment: 20+2 pages; major revision: new author, new structure, new result
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
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