Large-Scale Renormalisation of Fourier Transforms of Self-Similar Measures and Self-Similarity of Riesz Measures

We shall show that the oscillations observed by R. S. Strichartz in the Fourier transforms of self-similar measures have a large-scale renormalisation given by a Riesz measure. Vice versa the Riesz measure itself will be shown to be self-similar around every triadic point