Sums of reciprocals of fractional parts

2018/05/02 by Fregoli, Reynold
#FOS: Mathematics #Number Theory (math.NT)

paper · doi:10.48550/arxiv.1805.00865

Abstract

Let \boldsymbolα∈ ℝ^N and Q≥ 1. We consider the sum ∑_\boldsymbolq∈ [-Q,Q]^N∩ℤ^N\backslash\\boldsymbol0\‖\boldsymbolα⋅\boldsymbolq‖^-1. Sharp upper bounds are known when N=1, 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 N. Our result also sheds light on a question raised by Lê and Vaaler in 2013 on the sharpness of their lower bound ≫ Q^Nlog Q.

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