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arXiv · 1311.7277

Equivalence classes of codimension one cut-and-project nets

Alan Haynes
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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.

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Alan Haynes. 2013-11-28. Equivalence classes of codimension one cut-and-project nets. https://doi.org/10.1017/etds.2014.90

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