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In this paper, we prove new upper bounds for sums of reciprocals offractional parts over general aligned boxes, thus extending a previous resultof 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 ontheir diameter. This generalisation relies a novel lattice-point countingtechnique involving estimates for the higher successive minima of certainnaturally arising lattices.

Sums of Reciprocals of Fractional Parts over Aligned Boxes