Pure discrete spectrum and regular model sets in d-dimensional unimodular substitution tilings

Acta Crystallogr A Found Adv. 2020 Sep 1;76(Pt 5):600-610. doi: 10.1107/S2053273320009717. Epub 2020 Aug 21.


Primitive substitution tilings on {\bb R}^d whose expansion maps are unimodular are considered. It is assumed that all the eigenvalues of the expansion maps are algebraic conjugates with the same multiplicity. In this case, a cut-and-project scheme can be constructed with a Euclidean internal space. Under some additional condition, it is shown that if the substitution tiling has pure discrete spectrum, then the corresponding representative point sets are regular model sets in that cut-and-project scheme.

Keywords: Meyer sets; Pisot family substitution tilings; pure discrete spectrum; regular model sets; rigidity.