scholarly journals Lattice Bounded Distance Equivalence for 1D Delone Sets with Finite Local Complexity

2021 ◽  
Vol 59 ◽  
pp. 1-29
Author(s):  
Petr Ambroz ◽  
Zuzana Masakova ◽  
Edita Pelantova
Keyword(s):  
Finite Alphabet ◽  
Incidence Matrices ◽  
Project Method ◽  
Infinite Word ◽  
Bounded Distance ◽  
Local Complexity ◽  
Self Similar ◽  
Numeration Systems ◽  
Average Lattice ◽  
Delone Sets

Spectra of suitably chosen Pisot-Vijayaraghavan numbers represent non-trivial examples of self-similar Delone point sets of finite local complexity, indispensable in quasicrystal modeling. For the case of quadratic Pisot units we characterize, dependingly on digits in the corresponding numeration systems, the spectra which are bounded distance to an average lattice. Our method stems in interpretation of the spectra in the frame of the cut-and-project method. Such structures are coded by an infinite word over a finite alphabet which enables us to exploit combinatorial notions such as balancedness, substitutions and the spectrum of associated incidence matrices.

scholarly journals Number of bounded distance equivalence classes in hulls of repetitive Delone sets

2021 ◽  
Vol 0 (0) ◽  
pp. 0
Author(s):  
Dirk Frettlöh ◽  
Alexey Garber ◽  
Lorenzo Sadun
Keyword(s):  
Equivalence Class ◽  
Equivalence Relation ◽  
Equivalence Classes ◽  
Bounded Distance ◽  
Local Complexity ◽  
Delone Set ◽  
Delone Sets ◽  
Uniformly Bounded

<p style='text-indent:20px;'>Two Delone sets are bounded distance equivalent to each other if there is a bijection between them such that the distance of corresponding points is uniformly bounded. Bounded distance equivalence is an equivalence relation. We show that the hull of a repetitive Delone set with finite local complexity has either one equivalence class or uncountably many.</p>

scholarly journals SUBSTITUTIONS OVER INFINITE ALPHABET GENERATING (−β)-INTEGERS

2012 ◽  
Vol 23 (08) ◽  
pp. 1627-1639
Author(s):  
DANIEL DOMBEK
Keyword(s):  
Fixed Point ◽  
General Setting ◽  
Finite Alphabet ◽  
Infinite Word ◽  
Numeration Systems ◽  
Negative Base

We study positional numeration systems with negative base called (−β)-expansions in a more general setting than that of Ito and Sadahiro. We give an admissibility criterion for (−β)-expansions and discuss the properties of the set of (−β)-integers, denoted by ℤ−β. We give a description of distances between consecutive (−β)-integers and show that ℤ−β can be coded by an infinite word over an infinite alphabet, which is a fixed point of a non-erasing non-trivial morphism. We give a set of examples where ℤ−β is coded by an infinite word over a finite alphabet.

scholarly journals A dichotomy for bounded displacement equivalence of Delone sets

2021 ◽  
pp. 1-18
Author(s):  
YOTAM SMILANSKY ◽  
YAAR SOLOMON
Keyword(s):  
Compact Space ◽  
Equivalence Classes ◽  
Natural Topology ◽  
Fell Topology ◽  
Local Complexity ◽  
Substitution Tilings ◽  
Local Topology ◽  
Delone Sets

Abstract We prove that in every compact space of Delone sets in ${\mathbb {R}}^d$ , which is minimal with respect to the action by translations, either all Delone sets are uniformly spread or continuously many distinct bounded displacement equivalence classes are represented, none of which contains a lattice. The implied limits are taken with respect to the Chabauty–Fell topology, which is the natural topology on the space of closed subsets of ${\mathbb {R}}^d$ . This topology coincides with the standard local topology in the finite local complexity setting, and it follows that the dichotomy holds for all minimal spaces of Delone sets associated with well-studied constructions such as cut-and-project sets and substitution tilings, whether or not finite local complexity is assumed.

scholarly journals On substitution tilings and Delone sets without finite local complexity

2019 ◽  
Vol 39 (6) ◽  
pp. 3149-3177 ◽  
Author(s):  
Jeong-Yup Lee ◽  
◽  
Boris Solomyak ◽  
◽  
Keyword(s):  
Local Complexity ◽  
Substitution Tilings ◽  
Delone Sets

scholarly journals Defect Effect of Bi-infinite Words in the Two-element Case

2001 ◽  
Vol Vol. 4 no. 2 ◽  
Author(s):  
Ján Maňuch
Keyword(s):  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Finite Alphabet ◽  
Infinite Words ◽  
Infinite Word ◽  
The Family ◽  
International Audience ◽  
Necessary And Sufficient ◽  
Primitive Roots ◽  
Element Set

International audience Let X be a two-element set of words over a finite alphabet. If a bi-infinite word possesses two X-factorizations which are not shiftequivalent, then the primitive roots of the words in X are conjugates. Note, that this is a strict sharpening of a defect theorem for bi-infinite words stated in \emphKMP. Moreover, we prove that there is at most one bi-infinite word possessing two different X-factorizations and give a necessary and sufficient conditions on X for the existence of such a word. Finally, we prove that the family of sets X for which such a word exists is parameterizable.

scholarly journals Entropy ratio for infinite sequences with positive entropy

2018 ◽  
Vol 40 (3) ◽  
pp. 751-762 ◽  
Author(s):  
CHRISTIAN MAUDUIT ◽  
CARLOS GUSTAVO MOREIRA
Keyword(s):  
Exponential Growth ◽  
Fractal Dimensions ◽  
Combinatorial Structure ◽  
Finite Alphabet ◽  
Exponential Growth Rate ◽  
Positive Entropy ◽  
Infinite Words ◽  
Complexity Function ◽  
Infinite Word ◽  
Infinite Sequences

The complexity function of an infinite word $w$ on a finite alphabet $A$ is the sequence counting, for each non-negative $n$, the number of words of length $n$ on the alphabet $A$ that are factors of the infinite word $w$. For any given function $f$ with exponential growth, we introduced in [Complexity and fractal dimensions for infinite sequences with positive entropy. Commun. Contemp. Math. to appear] the notion of word entropy$E_{W}(f)$ associated to $f$ and we described the combinatorial structure of sets of infinite words with a complexity function bounded by $f$. The goal of this work is to give estimates on the word entropy $E_{W}(f)$ in terms of the limiting lower exponential growth rate of $f$.

Self-similar Delone sets and quasicrystals

1998 ◽  
Vol 31 (21) ◽  
pp. 4927-4946 ◽  
Author(s):  
Zuzana Masáková ◽  
Jirí Patera ◽  
Edita Pelantová
Keyword(s):  
Self Similar ◽  
Delone Sets

scholarly journals Local Complexity of Delone Sets and Crystallinity

2002 ◽  
Vol 45 (4) ◽  
pp. 634-652 ◽  
Author(s):  
Jeffrey C. Lagarias ◽  
Peter A. B. Pleasants
Keyword(s):  
Growth Rates ◽  
Linear Growth ◽  
Closed Ball ◽  
Covering Radius ◽  
Ideal Crystal ◽  
Minimum Radius ◽  
Local Complexity ◽  
Delone Set ◽  
Delone Sets

AbstractThis paper characterizes when a Delone set X in is an ideal crystal in terms of restrictions on the number of its local patches of a given size or on the heterogeneity of their distribution. For a Delone set X, let NX(T) count the number of translation-inequivalent patches of radius T in X and let MX(T) be the minimum radius such that every closed ball of radius MX(T) contains the center of a patch of every one of these kinds. We show that for each of these functions there is a “gap in the spectrum” of possible growth rates between being bounded and having linear growth, and that having sufficiently slow linear growth is equivalent to X being an ideal crystal.Explicitly, for NX(T), if R is the covering radius of X then either NX(T) is bounded or NX(T) ≥ T/2R for all T > 0. The constant 1/2R in this bound is best possible in all dimensions.For MX(T), either MX(T) is bounded or MX(T) ≥ T/3 for all T > 0. Examples show that the constant 1/3 in this bound cannot be replaced by any number exceeding 1/2. We also show that every aperiodic Delone set X has MX(T) ≥ c(n)T for all T > 0, for a certain constant c(n) which depends on the dimension n of X and is > 1/3 when n > 1.

Generalized Hausdorff dimensions of sets of real numbers with zero entropy expansion

2011 ◽  
Vol 32 (3) ◽  
pp. 1073-1089 ◽  
Author(s):  
CHRISTIAN MAUDUIT ◽  
CARLOS GUSTAVO MOREIRA
Keyword(s):  
Continued Fraction ◽  
Negative Integer ◽  
Finite Alphabet ◽  
Real Numbers ◽  
Continued Fraction Expansion ◽  
Hausdorff Dimensions ◽  
Subexponential Growth ◽  
Complexity Function ◽  
Infinite Word ◽  
Zero Entropy

AbstractThe complexity function of an infinite wordwon a finite alphabetAis the sequence counting, for each non-negative integern, the number of words of lengthnon the alphabetAthat are factors of the infinite wordw. Letfbe a given function with subexponential growth. The goal of this work is to estimate the generalized Hausdorff dimensions of the set of real numbers whoseq-adic expansion has a complexity function bounded byfand the set of real numbers whose continued fraction expansion is bounded byqand has a complexity function bounded byf.

scholarly journals Delone Sets with Finite Local Complexity: Linear Repetitivity Versus Positivity of Weights

2012 ◽  
Vol 49 (2) ◽  
pp. 335-347 ◽  
Author(s):  
Adnene Besbes ◽  
Michael Boshernitzan ◽  
Daniel Lenz
Keyword(s):  
Local Complexity ◽  
Delone Sets
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