# Periods and factors of weak model sets

## TL;DR

This paper investigates the dynamical systems associated with weak model sets, identifying their maximal equicontinuous factors and conditions for measure-theoretic isomorphisms, with applications to B-free systems.

## Contribution

It characterizes the maximal equicontinuous factors of weak model sets and establishes conditions for almost 1:1 extensions and measure-theoretic isomorphisms.

## Key findings

- Identified the maximal equicontinuous factor for certain weak model sets.
- Provided conditions under which the system is an almost 1:1 extension.
- Established measure-theoretic isomorphism to the Kronecker factor under specific conditions.

## 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 M\"obius disjointness conjecture. Here we continue our recent investigation [arXiv:1511.06137] of the extended hull ${\mathcal M}^{\scriptscriptstyle G}_{\scriptscriptstyle W}$, 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 ${\mathcal M}^{\scriptscriptstyle G}_{\scriptscriptstyle W}$ and give a sufficient condition when ${\mathcal M}^{\scriptscriptstyle G}_{\scriptscriptstyle W}$ 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 ${\mathcal M}^{\scriptscriptstyle G}_{\scriptscriptstyle W}$ equipped with its Mirsky measure is isomorphic to its Kronecker factor. For general nontrivial ergodic probability measures on ${\mathcal M}^{\scriptscriptstyle G}_{\scriptscriptstyle W}$, 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.