Higher-rank Bohr sets and multiplicative diophantine approximation

TL;DR

This paper advances the understanding of higher-rank Bohr sets and their role in multiplicative Diophantine approximation, extending Gallagher's theorem and solving a problem related to fibre refinement beyond the plane.

Contribution

It introduces a new structural theory of Bohr sets of arbitrary rank using reduced successive minima, generalizes results to inhomogeneous settings, and employs Diophantine transference inequalities instead of classical methods.

Findings

Developed a structural theory of higher-rank Bohr sets.

Extended Gallagher's theorem to inhomogeneous cases.

Solved a fibre refinement problem posed in 2015.

Abstract

Gallagher's theorem is a sharpening and extension of the Littlewood conjecture that holds for almost all tuples of real numbers. We provide a fibre refinement, solving a problem posed by Beresnevich, Haynes and Velani in 2015. Hitherto, this was only known on the plane, as previous approaches relied heavily on the theory of continued fractions. Using reduced successive minima in lieu of continued fractions, we develop the structural theory of Bohr sets of arbitrary rank, in the context of diophantine approximation. In addition, we generalise the theory and result to the inhomogeneous setting. To deal with this inhomogeneity, we employ diophantine transference inequalities in lieu of the three distance theorem.