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

Area of Science:

• Dynamical Systems
• Mathematical Physics
• Ergodic Theory

Background:

• Substitution tilings are fundamental in aperiodic order.
• Unimodular expansion maps and eigenvalue properties are crucial for tiling behavior.
• Cut-and-project schemes provide a framework for constructing model sets.

Purpose of the Study:

• To investigate the relationship between substitution tilings and model sets.
• To establish conditions under which substitution tilings correspond to regular model sets.
• To explore the implications of eigenvalue properties on tiling structures.

Main Methods:

• Analysis of primitive substitution tilings on Euclidean spaces (ℝᵈ).
• Consideration of unimodular expansion maps and their eigenvalues.
• Construction of cut-and-project schemes with Euclidean internal spaces.
• Application of spectral analysis (pure discrete spectrum).

Main Results:

• Demonstration that specific substitution tilings allow for the construction of cut-and-project schemes.
• Proof that under certain conditions, tilings with pure discrete spectra yield regular model sets.
• Establishment of a link between algebraic properties of eigenvalues and the geometric structure of tilings.

Conclusions:

• The study provides a theoretical framework connecting substitution tilings to regular model sets via cut-and-project schemes.
• Eigenvalue properties play a critical role in determining the regularity of point sets generated by substitution tilings.
• The findings contribute to the understanding of aperiodic order and its mathematical underpinnings.