  Tilings based on the cut and project method are key model systems for the
description of aperiodic solids. Typically, quantities of interest in
crystallography involve averaging over large patches, and are well defined only
in the infinite-volume limit. In particular, this is the case for
autocorrelation and diffraction measures. For cut and project systems, the
averaging can conveniently be transferred to internal space, which means
dealing with the corresponding windows. We illustrate this by the example of
averaged shelling numbers for the Fibonacci tiling and review the standard
approach to the diffraction for this example. Further, we discuss recent
developments for inflation-symmetric cut and project structures, which are
based on an internal counterpart of the renormalisation cocycle. Finally, we
briefly review the notion of hyperuniformity, which has recently gained
popularity, and its application to aperiodic structures.
