Stripes on rectangular tilings

Abstract

We consider a class of cut-and-project sets \Lambda = \Lambda_F \times \zahl in the plane. Let $L=\Lambda+w\real$, $w\in\real^2$, be a countable union of parallel lines. Then either (1) $L$ is a discrete family of lines, (2) $L$ is a dense subset of $\real^2$, or (3) each connected component of the closure of $L$ is homeomorphic to $[0,1] \times \real$.Comment: 6 pages, 2 figure