Littlewood and Duffin--Schaeffer-type problems in diophantine approximation

TL;DR

This paper extends Gallagher's theorem to inhomogeneous settings, introduces a diophantine fibre refinement, and establishes a sharp threshold for Liouville fibres, advancing understanding of multiplicative diophantine approximation.

Contribution

It provides the first inhomogeneous version of Gallagher's theorem and a new threshold for Liouville fibres, along with an inhomogeneous Duffin--Schaeffer conjecture for non-monotonic functions.

Findings

Established a fully-inhomogeneous Gallagher's theorem.

Proved a diophantine fibre refinement.

Identified a sharp threshold for Liouville fibres.

Abstract

Gallagher's theorem describes the multiplicative diophantine approximation rate of a typical vector. We establish a fully-inhomogeneous version of Gallagher's theorem, a diophantine fibre refinement, and a sharp and unexpected threshold for Liouville fibres. Along the way, we prove an inhomogeneous version of the Duffin--Schaeffer conjecture for a class of non-monotonic approximation functions.