Simultaneous Diophantine approximation - logarithmic improvements

TL;DR

This paper investigates a multiplicative Diophantine approximation problem, providing explicit rates for how closely certain products of affine forms can approach zero, using methods involving the density of higher-rank abelian group orbits.

Contribution

It offers the first explicit rate of approximation for Cassels' problem in multiplicative Diophantine approximation, utilizing a novel approach based on orbit density analysis.

Findings

Established explicit approximation rates for the product of affine forms.

Demonstrated the density of higher-rank abelian group orbits in this context.

Extended understanding of multiplicative Diophantine approximation behavior.

Abstract

This paper is devoted to the study of a problem of Cassels in multiplicative Diophantine approximation which involves minimising values of a product of affine linear forms computed at integral points. It was previously known that values of this product become arbitrary close to zero, and we establish that, in fact, they approximate zero with an explicit rate. Our approach is based on investigating quantitative density of orbits of higher-rank abelian groups.