Characterization of Cut-and-Project Sets Using a Binary Operation

Abstract

Cut-and-project sets with convex acceptance windows, based on irrationalities τ=\( \frac{1}{2} \)(1+√5), β=1+√2, μ=2+√3 are models for experimentally observed quasicrystals – materials with diffraction patterns consisting of sharp Bragg peaks in crystallographically disallowed patterns. We show that for each of these three irrationalities there exists a unique binary operation of the type x⊢sy:=sx+(1−s)y, such that one-dimensional cut-and-project sets are precisely Delone sets closed under this operation.