Pure discrete spectrum and regular model sets on some non-unimodular substitution tilings

Abstract

Substitution tilings with pure discrete spectrum are characterized as regular model sets whose cut-and-project scheme has an internal space that is a product of a Euclidean space and a profinite group. Assumptions made here are that the expansion map of the substitution is diagonalizable and its eigenvalues are all algebraically conjugate with the same multiplicity. A difference from the result of Lee et al. [Acta Cryst. (2020), A76, 600–610] is that unimodularity is no longer assumed in this paper.

Equivalence between pure point diffractive sets and cut-and-project sets on substitution tilings

“…Under the assumption of pure point diffraction spectrum, we can get a tiling S ∈ X T whose control point set C = (C j ) 1≤i≤4 fulfills C = lim n→∞ (Φ N ) n ({0}) and each point set C j is a regular model set with a window whose boundary measure is zero in the internal space R 2 . (see [11,12]).…”

“…where C = (C i ) i≤κ is a control point set of T . By choosing the tile map properly (see [12]), we can obtain a control point set C satisfying the following inclusion 3 + 2Z) using a proper tile-map, it is possible to describe the point sets as regular model sets (see [12,Example 5.11]).…”

“…The main result and the detailed proof is presented in [12]. In this manuscript, we look at the argument of the proof with a specific example.…”