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

Area of Science:

• Mathematical Physics
• Dynamical Systems
• Crystallography

Background:

• Substitution tilings are aperiodic structures with self-similarity properties.
• Discrete spectrum is a key characteristic for understanding the spectral properties of dynamical systems.
• Regular model sets provide a framework for studying ordered structures.

Purpose of the Study:

• To characterize substitution tilings with pure discrete spectrum.
• To investigate the role of the cut-and-project scheme's internal space.
• To relax the unimodularity assumption in previous characterizations.

Main Methods:

• Utilizing the theory of regular model sets.
• Analyzing the properties of the cut-and-project scheme.
• Applying concepts from group theory, specifically profinite groups.

Main Results:

• Substitution tilings with pure discrete spectrum correspond to regular model sets.
• The internal space of the cut-and-project scheme is a product of a Euclidean space and a profinite group.
• This characterization holds under assumptions of diagonalizable expansion maps with algebraically conjugate eigenvalues.

Conclusions:

• The study provides a refined characterization of substitution tilings with pure discrete spectrum.
• The findings extend previous results by removing the unimodularity condition.
• This work contributes to the understanding of aperiodic order in mathematical physics and crystallography.