Scale dependent partitioning of one-dimensional aperiodic set diffraction

Abstract

We give a multiresolution partition of pure point parts of diffraction patterns of one-dimensional aperiodic sets. When an aperiodic set is related to the Golden Ratio, denoted by $\tau$ , it is well known that the pure point part of its diffractive measure is supported by the extension ring of $\tau$ , denoted by $\mathbb{Z}[\tau]$ . The partition we give is based on the formalism of the so called $\tau$ -integers, denoted by $\mathbb{Z}_\tau$ . The set of $\tau$ -integers is a selfsimilar set obeying $\mathbb{Z}_\tau/\tau^{j-1}\subset\mathbb{Z}_\tau/\tau^j \subset \mathbb{Z}_\tau/\tau^{j + 1} \subset\mathbb{Z}[\tau]$ , $j\in\mathbb{Z}$ . The pure point spectrum is then partitioned with respect to this “Russian doll” like sequence of subsets $\mathbb{Z}_\tau/\tau^j$ . Thus we deduce the partition of the pure point part of the diffractive measure of aperiodic sets.