Classification and statistics of cut-and-project sets [PDF]
Journal of the European Mathematical Society (Print), 2020 We define Ratner-Marklof-Strombergsson measures. These are probability measures supported on cut-and-project sets in R^d (d>1) which are invariant and ergodic for the action of the groups ASL_d(R) or SL_d(R).
Ren'e Ruhr, Yotam Smilansky, B. Weiss
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Cut‐and‐project quasicrystals, lattices and dense forests [PDF]
Journal of the London Mathematical Society, 2019 Dense forests are discrete subsets of Euclidean space which are uniformly close to all sufficiently long line segments. The degree of density of a dense forest is measured by its visibility function.
F. Adiceam, Y. Solomon, B. Weiss
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Cut and project sets with polytopal window I: Complexity [PDF]
Ergodic Theory and Dynamical Systems, 2019 We calculate the growth rate of the complexity function for polytopal cut and project sets. This generalizes work of Julien where the almost canonical condition is assumed.
Henna Koivusalo, James J. Walton
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Relative Topological Entropy for Actions of Non-discrete Groups on Compact Spaces in the Context of Cut and Project Schemes [PDF]
Journal of Dynamics and Differential Equations, 2019 In the study of aperiodic order via dynamical methods, topological entropy is an important concept. In this paper, parts of the theory, like Bowen’s formula for fibre wise entropy or the independence of the definition from the choice of a Van Hove ...
T. Hauser
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Pisot substitution sequences, one dimensional cut-and-project sets and bounded remainder sets with fractal boundary [PDF]
Indagationes mathematicae, 2017 This paper uses a connection between bounded remainder sets in $\mathbb{R}^d$ and cut-and-project sets in $\mathbb{R}$ together with the fact that each one-dimensional Pisot substitution sequence is bounded distance equivalent to some lattice in order to
D. Frettloh, A. Garber
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Statistics of patterns in typical cut and project sets [PDF]
Ergodic Theory and Dynamical Systems, 2017 In this article pattern statistics of typical cubical cut and project sets are studied. We give estimates for the rate of convergence of appearances of patches to their asymptotic frequencies.
A. Haynes +3 more
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Gaps problems and frequencies of patches in cut and project sets [PDF]
Mathematical Proceedings of the Cambridge Philosophical Society, 2014 We establish a connection between gaps problems in Diophantine approximation and the frequency spectrum of patches in cut and project sets with special windows.
A. Haynes +3 more
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A short guide to pure point diffraction in cut-and-project sets [PDF]
, 2016 We briefly review the diffraction of quasicrystals and then give an elementary alternative proof of the diffraction formula for regular cut-and-project sets, which is based on Bochner’s theorem from Fourier analysis. This clarifies a common view that the
C. Richard, Nicolae Strungaru
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Equivalence classes of codimension-one cut-and-project nets [PDF]
Ergodic Theory and Dynamical Systems, 2013 We prove that in any totally irrational cut-and-project setup with codimension (internal space dimension) one, it is possible to choose sections (windows) in non-trivial ways so that the resulting sets are bounded displacement equivalent to lattices. Our
A. Haynes
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Constructing bounded remainder sets and cut-and-project sets which are bounded distance to lattices [PDF]
, 2014 For any irrational cut-and-project setup, we demonstrate a natural infinite family of windows which gives rise to separated nets that are each bounded distance to a lattice. Our proof provides a new construction, using a sufficient condition of Rauzy, of
A. Haynes, Henna Koivusalo
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