Badly approximable numbers and Littlewood-type problems

TL;DR

This paper proves that the set of pairs of real numbers with a specific Diophantine approximation property has full Hausdorff dimension, using a method adapted from Peres and Schlag to Littlewood-type problems.

Contribution

It introduces a novel application of the Peres-Schlag method to establish full Hausdorff dimension for a class of Littlewood-type problems involving badly approximable pairs.

Findings

The set of such pairs has full Hausdorff dimension in .

The method extends to various Littlewood-type problems.

Provides new insights into Diophantine approximation properties.

Abstract

We establish that the set of pairs $(\alpha, \beta)$ of real numbers such that $$ \liminf_{q \to + \infty} q \cdot (\log q)^2 \cdot \Vert q \alpha \Vert \cdot \Vert q \beta \Vert > 0, $$ where $\Vert \cdot \Vert$ denotes the distance to the nearest integer, has full Hausdorff dimension in $\R^2$. Our proof rests on a method introduced by Peres and Schlag, that we further apply to various Littlewood-type problems