A uniform metrical theorem in multiplicative Diophantine approximation

TL;DR

This paper establishes a uniform metrical theorem in multiplicative Diophantine approximation, providing an asymptotic count for the distribution of small product values involving two real numbers and integer multiples.

Contribution

It introduces a new asymptotic formula for counting how often the product of two fractional parts times an integer falls within specified bounds, under certain growth conditions.

Findings

Derived an asymptotic formula for the distribution of small product values

Established conditions under which the formula applies

Extended understanding of multiplicative Diophantine approximation in two dimensions

Abstract

For Lebesgue generic $(x_1,x_2)\in \mathbb{R}^2$, we investigate the distribution of small values of products $q\cdot \|qx_1\| \cdot \|qx_2\|$ with $q\in\mathbb{N}$, where $\|\cdot \|$ denotes the distance to the closest integer. The main result gives an asymptotic formula for the number of $1\le q\le T$ such that $$ a_T <q\cdot \|qx_1\| \cdot \|qx_2\|\leq b_T \quad \textrm{and} \quad \|qx_1\|, \|qx_2\|\leq c_T $$ for given sequences $a_T,b_T, c_T$ satisfying certain growth conditions.