Simultaneous diophantine approximation for a restricted class of pairs of real numbers

TL;DR

This paper proves the Littlewood conjecture for a specific class of badly approximable pairs of real numbers by analyzing roots of a cubic equation linked to their Diophantine properties.

Contribution

It introduces a novel approach using root localization of a cubic polynomial to establish the conjecture for restricted pairs.

Findings

Littlewood conjecture verified for certain badly approximable pairs

Root localization method effectively links Diophantine properties to polynomial roots

Estimates depend on continued fraction convergents of the pair

Abstract

We prove that the Littlewood conjecture is satisfied for a restricted class of pairs $(\alpha,\beta)$ of badly approximable numbers. We use the localization of the roots of a cubic equation with coefficients depending on the diophantine properties for the considered pair $(\alpha,\beta)$. The estimates of the roots rely on the properties of the denominators of the convergents of the continued fraction expansion of $\alpha$ and $\beta$.