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One-dimensional quasiperiodic tilings admitting progressions enclosure. (English. Russian original) Zbl 1195.11084
Russ. Math. 53, No. 7, 1-6 (2009); translation from Izv. Vyssh. Uchebn. Zaved., Mat. 2009, No. 7, 3-9 (2009).
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 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.

MSC:
11H06 Lattices and convex bodies (number-theoretic aspects)
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