Constructing bounded remainder sets and cut-and-project sets which are bounded distance to lattices

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 an infinite family of non-trivial bounded remainder sets for any totally irrational toral rotation in any dimension.Comment: 11 pages, 1 figure, updated references, changed intro to give credit to a result of Liardet which we were previously unaware o

Quantitative recurrence and the shrinking target problem for overlapping iterated function systems

In this paper we study quantitative recurrence and the shrinking target problem for dynamical systems coming from overlapping iterated function systems. Such iterated function systems have the important property that a point often has several distinct choices of forward orbit. As is demonstrated in this paper, this non-uniqueness leads to different behaviour to that observed in the traditional setting where every point has a unique forward orbit. We prove several almost sure results on the Lebesgue measure of the set of points satisfying a given recurrence rate, and on the Lebesgue measure of the set of points returning to a shrinking target infinitely often. In certain cases, when the Lebesgue measure is zero, we also obtain Hausdorff dimension bounds. One interesting aspect of our approach is that it allows us to handle targets that are not simply balls, but may have a more exotic geometry

Cut and project sets with polytopal window I: complexity

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. The analysis of polytopal cut and project sets has often relied on being able to replace acceptance domains of patterns by so-called cut regions. Our results correct mistakes in the literature where these two notions are incorrectly identified. One may only relate acceptance domains and cut regions when additional conditions on the cut and project set hold. We find a natural condition, called the quasicanonical condition, guaranteeing this property and demonstrate by counterexample that the almost canonical condition is not sufficient for this. We also discuss the relevance of this condition for the current techniques used to study the algebraic topology of polytopal cut and project sets