One-dimensional quasiperiodic tilings admitting progressions enclosure (Q735983)

Consider a tiling of the positive real line with two intervals that is based on an irrational rotation of the circle, and compare it with a lattice of the same point density as the set of left endpoints of the tiling intervals. A classic result by \textit{H. Kesten} [Acta Arith. 12, 193--212 (1966; Zbl 0144.28902)] gives a necessary and sufficient criterion for the existence of a bijection between the two point sets with bounded distance between images and pre-images. Here, the authors ask for the stronger property that each interval of the tiling contains precisely one lattice point in its interior (and no lattice point coincides with a boundary point). The lattice may be shifted for this purpose. Theorem 4 states necessary and sufficient conditions, while Theorems 5--9 give sufficient criteria that are perhaps more practical. The reader should exercise some care towards the details (e.g., \(\ell_1\neq \ell_2\) is implicit in Theorem 2, and equation (1) means the intersection of two open intervals), towards names (Ales in Ref. 1 should be Axel, and Rosy in Refs. 7 and 8 refers to Rauzy), and to the incompleteness of the list of references.