  Linearly repetitive cut and project sets are mathematical models for
perfectly ordered quasicrystals. In a previous paper we presented a
characterization of linearly repetitive cut and project sets. In this paper we
extend the classical definition of linear repetitivity to try to discover
whether or not there is a natural class of cut and project sets which are
models for quasicrystals which are better than `perfectly ordered'. In the
positive direction, we demonstrate an uncountable collection of such sets (in
fact, a collection with large Hausdorff dimension) for every choice of
dimension of the physical space. On the other hand we show that, for many
natural versions of the problems under consideration, the existence of these
sets turns out to be equivalent to the negation of a well known open problem in
Diophantine approximation, the Littlewood conjecture.
