Equivalence classes of codimension one cut-and-project nets

Abstract

We prove that in any totally irrational cut-and-project setup with codimension (internal space dimension) one, it is possible to choose sections (windows) in non-trivial ways so that the resulting sets are bounded displacement to lattices. Our proof demonstrates that for any irrational $\alpha$, regardless of Diophantine type, there is a collection of intervals in $\mathbb{R}/\mathbb{Z}$ which is closed under translation, contains intervals of arbitrarily small length, and along which the discrepancy of the sequence $\{n\alpha \}$ is bounded above uniformly by a constant.

19 pages, added some references and sharpened statements of some of the results

Cited by in corpus (4)

• Pisot substitution sequences, one dimensional cut-and-project sets and bounded remainder sets with fractal boundary
• Number of bounded distance equivalence classes in hulls of repetitive Delone sets
• Indistinguishable asymptotic pairs and multidimensional Sturmian configurations
• Weighted $1\times 1$ cut-and-project sets in bounded distance to a lattice