Mathematical quasicrystals with toric internal spaces, diffraction and rarefaction

Aperiodic crystals viewed as Delone sets of points on the real line, having an average lattice, are studied as congruence model λ-sets (physical space is Euclidean and the equivalent of the internal space is toric) in the context of cut-and-congruence λ-schemes, a new concept. When windows are finite point sets, the fractal rates of occupancy, at infinity, of the affine lattices associated with such Delone sets are shown to be simply related to the scaling exponents of the Fourier transform of the autocorrelation measure, completing results of Hof. These fractal rates of occupancy are named rarefaction laws. The case of the Thue–Morse quasicrystal, as a Meyer set, is explicitly developed. We present the arithmetics of 3-rarefaction phenomenon, the fractality of the Fourier transform of the autocorrelation measure. This new approach provides explicit formulae for singular continuous peaks and allows to discuss their possible extinction. In particular, this gives a possible sieve among Delone sets to be crystals in the new definition of a crystal by the IUCr in 1992. Ponctual scaling laws and the Bombieri–Taylor argument are considered.