Algorithm for determining pure pointedness of self-affine tilings

Abstract

Overlap coincidence in a self-affine tiling in R d is equivalent to pure point dynamical spectrum of the tiling dynamical system. We interpret the overlap coincidence in the setting of substitution Delone set in R d and find an efficient algorithm to check the pure point dynamical spectrum. This algorithm is easy to implement into a computer program. We give the program and apply it to several examples. In the course of the proof of the algorithm, we show a variant of the conjecture of Urbański (Solomyak (2006) [40]) on the Hausdorff dimension of the boundaries of fractal tiles.