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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.

52C20 Tilings in \(2\) dimensions (aspects of discrete geometry)
51M20 Polyhedra and polytopes; regular figures, division of spaces
Full Text: DOI
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