Asymptotic Diophantine approximation: The multiplicative case

TL;DR

This paper establishes precise asymptotic estimates for the count of integers satisfying a multiplicative Diophantine inequality involving two irrationals, providing new bounds and insights related to Littlewood's conjecture and fractional part sums.

Contribution

It offers a novel asymptotic estimate for the number of integers meeting a multiplicative fractional part condition, extending understanding in Diophantine approximation and related conjectures.

Findings

Derived asymptotics for the count of integers with multiplicative fractional parts

Provided new upper bounds on sums of reciprocals of fractional part products

Connected results to Littlewood's conjecture and recent open questions

Abstract

Let $\alpha$ and $\beta$ be irrational real numbers and $0<\F<1/30$. We prove a precise estimate for the number of positive integers $q\leq Q$ that satisfy $\|q\alpha\|\cdot\|q\beta\|<\F$. If we choose $\F$ as a function of $Q$ we get asymptotics as $Q$ gets large, provided $\F Q$ grows quickly enough in terms of the (multiplicative) Diophantine type of $(\alpha,\beta)$, e.g., if $(\alpha,\beta)$ is a counterexample to Littlewood's conjecture then we only need that $\F Q$ tends to infinity. Our result yields a new upper bound on sums of reciprocals of products of fractional parts, and sheds some light on a recent question of L\^{e} and Vaaler.