Periods and factors of weak model sets

Abstract

There is a renewed interest in weak model sets due to their connection to $\mathcal{B}$-free systems, which emerged from Sarnak's program on the Möbius disjointness conjecture. Here we continue our recent investigation [arXiv:1511.06137] of the extended hull ${\mathcal{M}}_W^G$, a dynamical system naturally associated to a weak model set in an abelian group $G$ with relatively compact window $W$. For windows having a nowhere dense boundary (this includes compact windows), we identify the maximal equicontinuous factor of ${\mathcal{M}}_W^G$ and give a sufficient condition when ${\mathcal{M}}_W^G$ is an almost 1:1 extension of its maximal equicontinuous factor. If the window is measurable with positive Haar measure and is almost compact, then the system ${\mathcal{M}}_W^G$ equipped with its Mirsky measure is isomorphic to its Kronecker factor. For general nontrivial ergodic probability measures on ${\mathcal{M}}_W^G$, we provide a kind of lower bound for the Kronecker factor. All relevant factor systems are natural $G$-actions on quotient subgroups of the torus underlying the weak model set. These are obtained by factoring out suitable window periods. Our results are specialised to the usual hull of the weak model set, and they are also interpreted for $\mathcal{B}$-free systems.

References in corpus (2)

• Pure point diffraction and cut and project schemes for measures: The smooth case
• Ergodic properties of visible lattice points

Cited by in corpus (8)

• On the Fourier Analysis of Measures with Meyer Set Support
• Modulated crystals and almost periodic measures
• On sampling and interpolation by model sets
• Pure point measures with sparse support and sparse Fourier--Bohr support
• Spectrum of weak model sets with Borel windows
• Why do Meyer sets diffract?
• Inter model sets in ${\mathbb{R}}^d$ are model sets
• Diffraction of the primes and other sets of zero density