"Understanding substitution tilings with pure discrete spectrum through a cut-and-project method"
Lee, Jeong-YupAfter the discovery of quasicrystal structures, there has been a lot of study on pure discrete spectrum of tiling dynamics as a characterizing property. There is a general theory that a regular model set, which is a cut and project set of a higher dimension lattice, has pure discrete spectrum [Schlottmann 2000]. But the converse is not true in general. We restrict tilings to substitution tilings and study the relation between pure discrete spectrum and regular model sets. Under certain assumptions on expansion maps of substitutions and ‘unimodularity’, the equivalence between the two notions has been shown [Lee-Akiyama-Lee, 2020]. In this talk, we eliminate the unimodularity condition in the earlier result and get the same equivalence.