# Bohr sets and multiplicative diophantine approximation

## TL;DR

This paper proves an inhomogeneous version of Gallagher's theorem in two dimensions, using Bohr sets and arithmetic progressions to verify conditions for Diophantine approximation results.

## Contribution

It introduces a new approach to inhomogeneous Diophantine approximation by analyzing Bohr sets and generalised arithmetic progressions, extending previous conditional results to unconditional ones.

## Key findings

- Established an inhomogeneous fibre version of Gallagher's theorem.
- Identified large generalized arithmetic progressions within Bohr sets.
- Verified Duffin--Schaeffer theorem hypotheses for the problem.

## Abstract

In two dimensions, Gallagher's theorem is a strengthening of the Littlewood conjecture that holds for almost all pairs of real numbers. We prove an inhomogeneous fibre version of Gallagher's theorem, sharpening and making unconditional a result recently obtained conditionally by Beresnevich, Haynes and Velani. The idea is to find large generalised arithmetic progressions within inhomogeneous Bohr sets, extending a construction given by Tao. This precise structure enables us to verify the hypotheses of the Duffin--Schaeffer theorem for the problem at hand, via the geometry of numbers.