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Induced bounded remainder sets. (English. Russian original) Zbl 1372.52024
St. Petersbg. Math. J. 28, No. 5, 671-688 (2017); translation from Algebra Anal. 28, No. 5, 171-194 (2017).
Summary: The induced two-dimensional Rauzy tilings are generalized to tiling of the tori $$\mathbb{T}^D= \mathbb{R}^D/ \mathbb{Z}^D$$ of arbitrary dimension $$D$$. For that, a technique of embedding $$T \mathop{\hookrightarrow}\limits^{\mathrm{em}} \mathbb{T}^D$$ of toric developments $$T$$ into the torus $$\mathbb{T}^D_L = \mathbb{R}^D/ L$$ for some lattice $$L$$ is used. A feature of the developments $$T$$ is that for a given shift $$S: \mathbb{T}^D \longrightarrow \mathbb{T}^D$$ of the torus, its restriction $$S| _T$$ to the subset $$T \subset \mathbb{T}^D$$, i.e., the first recurrence map, or the Poincaré map, is equivalent to an exchange transformation of the tiles $$T_k$$ that form a tiling of the development $$T=T_0\sqcup T_1\sqcup \dots \sqcup T_D$$. In the case under consideration, the induced map $$S| _T$$ is a translation of the torus $$\mathbb{T}^D_L$$. It is proved that every $$T_k$$ is a bounded remainder set: the deviations $$\delta _{T_k}(i,x_{0})$$ in the formula $$r_{T_k}(i,x_{0})= a_{T_k} i + \delta _{T_k}(i,x_{0})$$ are bounded, where $$r_{T}(i,x_{0})$$ is the number of occurrences of the points $$S^{0}(x_{0}), S^{1}(x_{0}),\dots , S^{i-1}(x_{0})$$ from the $$S$$-orbit in the set $$T_k$$, $$x_0$$ is an arbitrary starting point on the torus $$\mathbb{T}^D$$, and the coefficient $$a_{T_k}$$ equals the volume of $$T_k$$. Explicit estimates are obtained for these deviations $$\delta _{T_k}(i,x_{0})$$. Earlier, the relationship between the maps $$S| _T$$ and bounded remainder sets was noticed by Rauzy and Ferenczi.

##### MSC:
 52C20 Tilings in $$2$$ dimensions (aspects of discrete geometry) 51M20 Polyhedra and polytopes; regular figures, division of spaces
##### Keywords:
Poincaré map; bounded remainder sets
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##### References:
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