Induced bounded remainder sets

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 $T\stackrel {\operatorname {em}}{\hookrightarrow } \mathbb {T}^D$ 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.