  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.
