  The cut and project method is a central construction in the theory of
Aperiodic Order for generating quasicrystals with pure point diffraction.
Linear repetitivity ({\bf LR}) is a form of ideal regularity of aperiodic
patterns. Recently, Koivusalo and the present author characterised {\bf LR} for
cut and project sets with convex polytopal windows whose supporting hyperplanes
are commensurate with the lattice, the weak homogeneity property. For such cut
and project sets, we show that {\bf LR} is equivalent to two properties. One is
a low complexity condition, which may be determined from the cut and project
data by calculating the ranks of the intersections of the projection of the
lattice to the internal space with the subspaces parallel to the supporting
hyperplanes of the window. The second condition is that the projection of the
lattice to the internal space is Diophantine (or `badly approximable'), which
loosely speaking means that the lattice points in the total space stay far from
the physical space, relative to their norm. We review then extend these results
to non-convex and disconnected polytopal windows, as well as windows with
polytopal partitions producing cut and project sets of labelled points.
Moreover, we obtain a complete characterisation of {\bf LR} in the fully
general case, where weak homogeneity is not assumed. Here, the Diophantine
property must be replaced with an inhomogeneous analogue. We show that cut and
project schemes with internal space isomorphic to \(\mathbb{R}^n \oplus G
\oplus \mathbb{Z}^r\), for \(G\) finite Abelian, can, up to MLD equivalence, be
reduced to ones with internal space \(\mathbb{R}^n\), so our results also cover
cut and project sets of this form, such as the (generalised) Penrose tilings.
