On pattern entropy of weak model sets

Abstract

We study point sets arising from cut-and-project constructions. An important class is weak model sets, which include squarefree numbers and visible lattice points. For such model sets, we give a non-trivial upper bound on their pattern entropy in terms of the volume of the window boundary in internal space. This proves a conjecture by R.V. Moody.

14 pages; v3: changes in some proofs, v4: minor changes in some proofs

References in corpus (5)

• Quasicrystals and Poisson's summation formula
• Pure point diffraction and cut and project schemes for measures: The smooth case
• Ergodic properties of visible lattice points
• Pure point diffraction implies zero entropy for Delone sets with uniform cluster frequencies
• On Weighted Dirac Combs Supported Inside Model Sets

Cited by in corpus (16)

• Dynamics on the graph of the torus parametrisation
• Pure Point Diffraction and Mean, Besicovitch and Weyl Almost Periodicity
• Aperiodic order and spherical diffraction, I: Auto-correlation of model sets
• A short guide to pure point diffraction in cut-and-project sets
• Toeplitz flows and model sets
• Periods and factors of weak model sets
• Leptin densities in amenable groups
• Fourier transform of Rauzy fractals and point spectrum of 1D Pisot inflation tilings
• Model sets with positive entropy in Euclidean cut and project schemes
• On sampling and interpolation by model sets
• Model sets with precompact Borel windows
• Pure point measures with sparse support and sparse Fourier--Bohr support
• Spectrum of weak model sets with Borel windows
• On the Garden of Eden theorem for B-free subshifts
• Dynamical spectrum of power-free integers in quadratic number fields and beyond
• Amorphic complexity of group actions with applications to quasicrystals