Metrical musings on Littlewood and friends

TL;DR

This paper proves a metrical result showing that a large set of badly approximable numbers satisfy a strong version of several Littlewood-related conjectures, expanding understanding of their measure-theoretic properties.

Contribution

It establishes that the set of numbers satisfying strong versions of multiple Littlewood-related conjectures has full Hausdorff dimension within badly approximable numbers.

Findings

Set of numbers satisfying strong conjecture versions is large in Hausdorff dimension.

Results connect Littlewood conjecture variants with measure-theoretic size.

Advances understanding of the distribution of solutions in Diophantine approximation.

Abstract

We prove a metrical result on a family of conjectures related to the Littlewood conjecture, namely the original Littlewood conjecture, the mixed Littlewood conjecture of de Mathan and Teuli\'e and a hybrid between a conjecture of Cassels and the Littlewood conjecture. It is shown that the set of numbers satisfying a strong version of all of these conjectures is large in the sense of Hausdorff dimension restricted to the set of badly approximable numbers.