Sums of reciprocals of fractional parts

TL;DR

This paper develops a new method to establish sharp upper bounds for sums of reciprocals of fractional parts in higher dimensions, extending known results from the one-dimensional case.

Contribution

It introduces a novel approach based on lattice counting to derive sharp bounds for sums involving fractional parts in arbitrary dimensions.

Findings

Established sharp upper bounds for sums of reciprocals of fractional parts in any dimension.

Extended techniques beyond continued fractions and three distance theorem.

Provided insights into the sharpness of lower bounds posed by Lê and Vaaler.

Abstract

Let $\boldsymbol{\alpha}\in \mathbb{R}^N$ and $Q\geq 1$. We consider the sum $\sum_{\boldsymbol{q}\in [-Q,Q]^N\cap\mathbb{Z}^N\backslash\{\boldsymbol{0}\}}\|\boldsymbol{\alpha}\cdot\boldsymbol{q}\|^{-1}$. 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 L\^{e} and Vaaler in 2013 on the sharpness of their lower bound $\gg Q^N\log Q$.