Sets of bounded discrepancy for multi-dimensional irrational rotation
Sets of bounded discrepancy for multi-dimensional irrational rotation
复制标题
多维无理旋转的有界差异集
DOI：
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发表时间：
2014
期刊：
影响因子：
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通讯作者：
Nir Lev
中科院分区：
文献类型：
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作者：
Sigrid Grepstad;Nir Lev
我们研究关于 d 维环面无理旋转的有界余数集。这个主题可以追溯到 Hecke、Ostrowski 和 Kesten，他们用一维有界余数来表征区间。首先，我们通过构造一类有界余数的 d 维平行六面体，将 Hecke-Ostrowski 结果扩展到多个维度。然后，我们用这种平行六面体的“等可分解性”来表征黎曼可测有界余数集。通过构造关于这种等分解的不变量，我们导出了多胞形成为有界余数集的显式条件。特别是，这产生了二维凸有界余数多边形的表征。该方法还用于获得其他几个结果。
We study bounded remainder sets with respect to an irrational rotation of the d-dimensional torus. The subject goes back to Hecke, Ostrowski and Kesten who characterized the intervals with bounded remainder in dimension one.First we extend to several dimensions the Hecke–Ostrowski result by constructing a class of d-dimensional parallelepipeds of bounded remainder. Then we characterize the Riemann measurable bounded remainder sets in terms of “equidecomposability” to such a parallelepiped. By constructing invariants with respect to this equidecomposition, we derive explicit conditions for a polytope to be a bounded remainder set. In particular this yields a characterization of the convex bounded remainder polygons in two dimensions. The approach is used to obtain several other results as well.