Sets of bounded discrepancy for multi-dimensional irrational rotation

TL;DR

This paper extends the theory of bounded remainder sets from one dimension to multiple dimensions, characterizing such sets for irrational rotations on the torus using geometric and invariance methods.

Contribution

It introduces a class of multi-dimensional parallelepipeds of bounded remainder and characterizes bounded remainder sets via equidecomposability and invariants.

Findings

Constructed multi-dimensional parallelepipeds of bounded remainder

Characterized bounded remainder sets through equidecomposability

Provided explicit conditions for convex bounded remainder polygons in two dimensions

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.