Consequences of Pure Point Diffraction Spectra for Multiset Substitution Systems

Abstract

There is a growing body of results in the theory of discrete point sets and tiling systems giving conditions under which such systems are pure point diffractive. Here we look at the opposite direction: what can we infer about a discrete point set or tiling, defined through a primitive substitution system, given that it is pure point diffractive? Our basic objects are Delone multisets and tilings, which are self-replicating under a primitive substitution system of affine mappings with a common expansive map $Q$. Our first result gives a partial answer to a question of Lagarias and Wang: we characterize repetitive substitution Delone multisets that can be represented by substitution tilings using a concept of "legal cluster". This allows us to move freely between both types of objects. Our main result is that for lattice substitution multiset systems (in arbitrary dimensions) being a regular model set is not only sufficient for having pure point spectrum--a known fact--but is also necessary. This completes a circle of equivalences relating pure point dynamical and diffraction spectra, modular coincidence, and model sets for lattice substitution systems begun by the first two authors of this paper.

36 pages

Cited by in corpus (11)

• Spectral and topological properties of a family of generalised Thue-Morse sequences
• Substitution Delone Sets with Pure Point Spectrum are Inter Model Sets
• Limit theorems for self-similar tilings
• Aperiodic order and pure point diffraction
• Squirals and beyond: Substitution tilings with singular continuous spectrum
• Overlap coincidence to strong coincidence in substitution tiling dynamics
• The Pisot Conjecture for $\beta$-substitutions
• Multiscale Substitution Tilings
• On arithmetic progressions in non-periodic self-affine tilings
• Discrete quasiperiodic sets with predefined local structure
• Decidability of the isomorphism problem between multidimensional substitutive subshifts