  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.
