  In this paper, we prove new upper bounds for sums of reciprocals of
fractional parts over general aligned boxes, thus extending a previous result
of the author concerning bounds for sums of reciprocals over symmetric boxes.
These new upper bounds depend solely on the volume of the boxes, and not on
their diameter. This generalisation relies a novel lattice-point counting
technique involving estimates for the higher successive minima of certain
naturally arising lattices.
