Fully Inhomogeneous Multiplicative Diophantine Approximation of Badly Approximable Numbers

TL;DR

This paper advances the understanding of inhomogeneous multiplicative Diophantine approximation for badly approximable numbers, proving new results that extend classical conjectures and generalize recent findings.

Contribution

It establishes a strong form of Littlewood's conjecture with inhomogeneous shifts for a full-dimensional set and generalizes previous results to inhomogeneously badly approximable numbers.

Findings

Proved a strong inhomogeneous Littlewood-type conjecture for a full-dimensional set.

Generalized results to inhomogeneous badly approximable numbers.

Made progress on a problem posed by Pollington, Velani, Zafeiropoulos, and Zorin.

Abstract

We establish a strong form of Littlewood's conjecture with inhomogeneous shifts, for a full-dimensional set of pairs of badly approximable numbers on a vertical line. We also prove a uniform assertion of this nature, generalising a strong form of a result by Haynes, Jensen and Kristensen. Finally, we establish a similar result involving inhomogeneously badly approximable numbers, making progress towards a problem posed by Pollington, Velani, Zafeiropoulos and Zorin.