2017/02/08 by Stanisław Kasjan, Gerhard Keller, Mariusz Lemańczyk · 1 citation
Mathematics · #math.DS #msc:37A35 #msc:37A45 #msc:37B05
paper · pdf · doi:10.1093/imrn/rnx196
arxiv created 2017/02/08 · arxiv updated 2019/05/16
Let \mathcal B be an infinite subset of \1,2,…\. We characterize arithmetic and dynamical properties of the \mathcal B-free set \mathcal F\mathcal B through group theoretical, topological and measure theoretic properties of a set W (called the window) associated with \mathcal B. This point of view stems from the interpretation of the set \mathcal F\mathcal B as a weak model set. Our main results are: \mathcal B is taut if and only if the window is Haar regular; the dynamical system associated to \mathcal F\mathcal B is a Toeplitz system if and only if the window is topologically regular; the dynamical system associated to \mathcal F\mathcal B is proximal if and only if the window has empty interior; and the dynamical system associated to \mathcal F\mathcal B has the "naïvely expected" maximal equicontinuous factor if and only if the interior of the window is aperiodic.