Sums of reciprocals of fractional parts

Abstract

Let [Formula: see text] and [Formula: see text]. We consider the sum [Formula: see text]. Sharp upper bounds are known when [Formula: see text], using continued fractions or the three-distance theorem. However, these techniques do not seem to apply in higher dimension. We introduce a different approach, based on a general counting result of Widmer for weakly admissible lattices, to establish sharp upper bounds for arbitrary [Formula: see text]. Our result also sheds light on a question raised by Lê and Vaaler in 2013 on the sharpness of their lower bound [Formula: see text].