Sets of bounded discrepancy for multi-dimensional irrational rotation

2014/04/01 by Sigrid Grepstad, Nir Lev, Grepstad, Sigrid +1 · 1 citation
Mathematics · #11J71 #11K38 #52B45 #Analytic Number Theory Research #Dynamical Systems (math.DS) #FOS: Mathematics #Mathematical Approximation and Integration #Mathematics and Applications #Number Theory (math.NT) #math.DS #math.NT #msc:11J71 #msc:11K38 #msc:52B45

paper · pdf · doi:10.48550/arxiv.1404.0165

To appear in Geometric And Functional Analysis

openalex publication_date 2014/04/01 · arxiv created 2014/10/22 · arxiv updated 2014/10/23 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

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.

Cited by

• Bounded remainder sets, bounded distance equivalent cut-and-project sets, and equidecomposability

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