Badly approximable numbers related to the Littlewood conjecture

TL;DR

This paper demonstrates the existence of real number pairs with specific approximation properties related to the Littlewood conjecture, using Peres-Schlag's method to establish a positive lower bound for a certain liminf expression.

Contribution

It introduces a novel application of Peres-Schlag's method to establish the existence of real numbers with bounded approximation behavior related to the Littlewood conjecture.

Findings

Existence of real numbers with positive liminf of q log^2 q times the product of fractional parts.

Application of Peres-Schlag's method to a new problem in Diophantine approximation.

Provides a lower bound for the approximation liminf involving two real numbers.

Abstract

By means of Peres-Schlag's method we prove the existence of real numbers $\alpha, \beta$ such that $$ \liminf_{q\to \infty} (q\log^2 q)||\alpha q|| ||\beta q|| > 0.