PURE DISCRETE SPECTRUM AND REGULAR MODEL SETS IN UNIMODULAR SUBSTITUTION TILINGS ON Rd

The paper proves that, under a diagonalizable unimodular expansion whose eigenvalues are algebraic conjugates of equal multiplicity, a primitive repetitive substitution tiling has pure discrete spectrum if and only if each control-point set is a regular Euclidean model set for a canonical CPS. It constructs the Euclidean CPS from conjugates, proves equality Ci = Λ(Ui) with Ui open, and establishes boundary measure 0 for the windows, then invokes known regular-model-set ⇒ PDS results to conclude the equivalence. The candidate solution outlines the same equivalence with a different emphasis: it appeals to overlap/algebraic coincidence and Rauzy-type graph-directed windows, and to standard model-set dynamics plus MLD. Its logic is broadly sound, though it overstates disjointness of window interiors and tacitly assumes rigidity that the paper derives. Overall, both are correct, via different proof routes.