PPubMed12 Aug 2022Cited 1×

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

Jeong-Yup Lee, M. I. Aroyo

Abstract

The equivalence between pure discrete spectrum and regular model sets on some non-unimodular substitution tilings is established. This will help to provide useful information about the cut-and-project scheme used in the description of quasiperiodic structures.

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The paper proves that for certain non‑unimodular substitution tilings, pure discrete spectrum is equivalent to the control point sets being regular model sets in a cut‑and‑project scheme.

What’s new

i.Establishes equivalence between pure discrete spectrum and regular model sets for non‑unimodular substitution tilings
ii.Constructs a cut‑and‑project scheme with an internal space that is a product of Euclidean and profinite groups
iii.Shows control point sets are regular model sets with windows of Haar measure zero
iv.Extends rigidity structure results to non‑unimodular cases
v.Provides concrete examples illustrating the theory

How they did it

The authors first outline the proof under DAMR assumptions for simple substitution tilings, then define an appropriate internal space and construct a cut‑and‑project scheme (CPS). Using the algebraic coincidence property, they demonstrate that control point sets become regular model sets, and finally prove the windows have zero Haar measure.

What they found

Theorem 1.1: Pure discrete spectrum ⇒ each control point set is a regular model set in a CPS with Euclidean × profinite internal space
Corollary 5.9: Pure discrete spectrum ⇔ each control point set is a regular model set in such a CPS
Proposition 5.3: Under CPS existence, pure discrete spectrum yields Meyer sets and model‑set structure
Proposition 5.6: Windows of the model sets have Haar measure zero, establishing regularity

Caveats

✕Rigidity property is proved only for diagonalizable expansion maps with eigenvalues algebraically conjugate and equal multiplicity
✕Results rely on DAMR assumptions; broader substitution tilings are not covered
✕Extension to non‑rigid or more general tilings remains an open question

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