GCD sums and sum-product estimates

TL;DR

This paper introduces a new estimate for GCD sums that enhances previous bounds and applies it to improve results on sequence equidistribution modulo 1, especially for subsets of squares being metric poissonian.

Contribution

It provides a novel bound on GCD sums and demonstrates its application in advancing the understanding of sequence distribution and metric poissonian properties.

Findings

Improved GCD sum estimates over previous bounds.

Enhanced results on the equidistribution of sequences modulo 1.

Proved that arbitrary subsets of squares are metric poissonian.

Abstract

In this note we prove a new estimate on so-called GCD sums (also called G\'{a}l sums), which, for certain coefficients, improves significantly over the general bound due to de la Bret\`{e}che and Tenenbaum. We use our estimate to prove new results on the equidistribution of sequences modulo 1, improving over a result of Aistleitner, Larcher, and Lewko on how the metric poissonian property relates to the notion of additive energy. In particular, we show that arbitrary subsets of the squares are metric poissonian.