Arithmetic properties of spectra produced by Farey hierarchies of approximants

摘要

We discuss the consequences of the hierarchical nature of series of approximants of aperiodic crystals on their diffraction patterns and spectra of elementary excitations. We show how a linear form defined on Z~3 can be used to order Bragg reflections in diffraction patterns according to their amplitudes, and gaps in spectra of elementary excitations according to their widths, for all the structures in the hierarchy. Bragg peaks amplitudes and gap widths are projective functions on P(Z~3), recursively defined on 2D Farey sets (generalization of Farey series).