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Exchanged toric developments and bounded remainder sets. (English. Russian original) Zbl 1272.11087
J. Math. Sci., New York 184, No. 6, 716-745 (2012); translation from Zap. Nauchn. Semin. POMI 392, 95-145 (2011).
The paper concerns the distribution of fractional parts of integer multiples of vectors in $$\mathbb{R}^D$$.
Depending on a vector $$s \in \mathbb{R}^D$$, the author defines polyhedra $$F^s \subseteq \mathbb{R}^D$$, so-called stretched or mirror-stretched cubes, that give rise to a tiling $$\{F^s+l: l \in L\}$$ of $$\mathbb{R}^D$$ based on a lattice $$L=L(s) \subseteq \mathbb{R}^D$$. Another parameter $$0 < \lambda < 1$$ is used for introducing a particular congruence by dissection of $$F^s$$ with itself that is realized by a dissection of $$F^s$$ into polyhedra $$F^{s,\lambda}_k$$, $$k=0,\ldots,D$$. The map $$x \mapsto x\, \mathrm{mod}\, L$$ of $$F^s$$ onto the torus $$\mathbb{T}^s = \mathbb{R}^D/L$$ induces a related dissection of $$\mathbb{T}^s$$ into $$\mathbb{T}^{s,\lambda}_k$$, $$k=0,\ldots,D$$. If the parameters $$s$$ and $$\lambda$$ satisfy a condition of irrationality, the sets $$\mathbb{T}^{s,\lambda}_k$$ show the behavior of bounded remainder sets: for every $$n \in \{1,2,\ldots\}$$ and every $$l \in L$$, the shift vector $$\beta=\frac{\lambda s + l}{n}$$ satisfies $\left|\{j: j \beta \,\mathrm{mod}\,L \in \mathbb{T}^{s,\lambda}_k, 0 \leq j < i\}| - i \frac{\mathrm{vol}(\mathbb{T}^{s,\lambda}_k)}{\mathrm{vol}(\mathbb{T}^s)} \right| \leq c(s,k) n$ for $$i=0,1,2,\ldots$$ Similar consequences are obtained for certain kinds of dissections of the above tilings of tori.

##### MSC:
 11J71 Distribution modulo one 52B45 Dissections and valuations (Hilbert’s third problem, etc.) 52C07 Lattices and convex bodies in $$n$$ dimensions (aspects of discrete geometry) 52C22 Tilings in $$n$$ dimensions (aspects of discrete geometry)
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##### References:
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