Algorithm for determining pure pointedness of self-affine tilings

Overlap coincidence in a self-affine tiling in Rd 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 Rd 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.