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

A characterisation of linear repetitivity for cut and project sets with general polytopal windows