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

Abstract

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.

Pattern Equivariant Mass Transport in Aperiodic Tilings and Cohomology

“…There has been a burst of activity in recent years studying the bounded displacement (BD) equivalence relation for tilings and Delone sets [16,19,17,18,24,1,39,38]. As we reported in our previous paper [24], there are many relevant papers contributing to the subject that predate the terminology BD such as [7,8,26].…”

“…A closely related notion to BD is that of a bounded remainder set (BRS). There has been quite a bit of activity (including new cohomological work) in this subject area in recent years [14,13,12,19,17,27,24].…”

Open Problems and Conjectures Related to the Theory of Mathematical Quasicrystals

“…In Haynes et al (2014), a more general result is proved, which applies for all choices of k and d with a Diophantine hypothesis on E, and for all windows W with the property that the upper Minkowski dimension of ∂W is less than k − d. Further results related to this problem can be found in Haynes (2016) and Haynes and Koivusalo (2016).…”

Bounded Remainder Sets on a Double Covering of the Klein Bottle

“…The structure of bounded remainder sets X on tori M = T D = R D /Z D of arbitrary dimensions D is understood sufficiently well [1][2][3][4][5][6][7]. Moreover, at present after a rather long pause [8][9][10][11] new ideas have appeared due to an extensive penetration of quasilattices and quasicrystals into different fields of mathematics and physics, and the interest in studying multidimensional toric bounded remainder sets has reappeared (see [6,7]).…”

“…Moreover, at present after a rather long pause [8][9][10][11] new ideas have appeared due to an extensive penetration of quasilattices and quasicrystals into different fields of mathematics and physics, and the interest in studying multidimensional toric bounded remainder sets has reappeared (see [6,7]). For instance, in [2][3][4][5] a general method for constructing such sets X was developed.…”