Modulated quasicrystals

There is some confusion in the literature what "modulated quasicrystals" are: Some people treat "modulated quasicrystals" and "deformed model sets" as exchangeable termini (compare [6, 9, 5]), others claim that "[...] the projection method becomes powerless against incommensurate modulated structures" (e.g., [12 p. 148]). We use a mathematical approach and propose the following classification: While deformed model sets are characterised by a deformation that does not change the location of the Bragg peaks, a modulated phase yields "satellites" in the diffraction pattern and is achieved "by enlarging the internal space" with a torus.