Constraints on pure point diffraction on aperiodic point patterns of finite local complexity

TL;DR

This paper establishes linear constraints linking the partial amplitudes of the pure point diffraction spectrum for aperiodic Delone point patterns with finite local complexity, derived from the tiling's geometric structure.

Contribution

It introduces explicit linear relations among diffraction amplitudes based on the geometry of the prototile space for aperiodic patterns.

Findings

Partial amplitudes are linked by linear constraints.

Constraints are explicitly derived from tiling geometry.

Results apply to aperiodic Delone point patterns with finite local complexity.

Abstract

It is shown that the partial amplitudes of the pure point part of the diffraction spectrum of an aperiodic Delone point pattern of finite local complexity are linked by a set of linear constraints. These relations can be explicitly derived from the geometry of the prototile space of the underlying tiling.