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arXiv · 1404.0165

Sets of bounded discrepancy for multi-dimensional irrational rotation

Sigrid Grepstad·Nir Lev
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Abstract

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.

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Sigrid Grepstad, Nir Lev. 2014-04-01. Sets of bounded discrepancy for multi-dimensional irrational rotation. https://arxiv.org/abs/1404.0165

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