  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.
