Equivalence classes of codimension-one cut-and-project nets

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 equivalent to lattices. Our proof demonstrates that for any irrational alpha, regardless of Diophantine type, there is a collection of intervals in R/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.