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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 whocharacterized the intervals with bounded remainder in dimension one.  First we extend to several dimensions the Hecke-Ostrowski result byconstructing 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 withrespect to this equidecomposition, we derive explicit conditions for a polytopeto be a bounded remainder set. In particular this yields a characterization ofthe convex bounded remainder polygons in two dimensions. The approach is usedto obtain several other results as well.

Sets of bounded discrepancy for multi-dimensional irrational rotation