Diffraction of the pinwheel tiling

Determine the natural diffraction measure of the Conway–Radin pinwheel tiling by computing the Fourier transform of the autocorrelation of the Dirac comb associated with the tiling.

Background

The paper studies spherically symmetric diffraction concentrated on a single sphere and answers a question of Strungaru by showing that spherical waves have diffraction supported on a single sphere. In motivating this, the authors reference tilings with statistical circular symmetry, highlighting the pinwheel tiling as a prominent example.

They explicitly note that the diffraction of the pinwheel tiling remains unresolved. This long-standing problem asks for the precise determination of the diffraction associated with the pinwheel tiling, which is expected to exhibit circular symmetry but whose exact measure has not been established.