  One of the simplest non-Pisot substitution rules is investigated in its
geometric version as a tiling with intervals of natural length as prototiles.
Via a detailed renormalisation analysis of the pair correlation functions, we
show that the diffraction measure cannot comprise any absolutely continuous
component. This implies that the diffraction, apart from a trivial Bragg peak
at the origin, is purely singular continuous. En route, we derive various
geometric and algebraic properties of the underlying Delone dynamical system,
which we expect to be relevant in other such systems as well.
