Exchanged toric developments and bounded remainder sets
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.
fractional part with respect to a lattice
scientific article; zbMATH DE number 6111011