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

Abstract

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.

11 pages, 1 figure, updated references, changed intro to give credit to a result of Liardet which we were previously unaware of