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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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