Cut and project sets with polytopal window I: Complexity

Abstract

We calculate the growth rate of the complexity function for polytopal cut and project sets. This generalises work of Julien where the almost canonical condition is assumed. The analysis of polytopal cut and project sets has often relied on being able to replace acceptance domains of patterns by so-called cut regions. Our results correct mistakes in the literature where these two notions are incorrectly identified. One may only relate acceptance domains and cut regions when additional conditions on the cut and project set hold. We find a natural condition, called the quasicanonical condition, guaranteeing this property and demonstrate via counterexample that the almost canonical condition is not sufficient for this. We also discuss the relevance of this condition for the current techniques used to study the algebraic topology of polytopal cut and project sets.

Complexity of Cut-and-Project Sets of Polytopal Type in Special Homogeneous Lie Groups

“…In particular, since we want to follow the approach of Julien, [26], and Koivusalo and Walton, [27], we need a notion of hyperplanes. So a first question is, in which groups can we define hyperplanes?…”

“…First we will establish the connection between the equivalence classes of patches and the acceptance domains in Section 2. This is a translation from the Euclidean case considered in [27]. The only difference is that we have to be a bit more careful since our groups are in general non-abelian.…”

“…In the third step we show in Section 5 that we can estimate the number of acceptance domains by extending the boundary of the window. This is also done in [27], the new regions inside the window are called cut regions in this paper. Again the result we obtain is the same as in the Euclidean case.…”

“…Optimal results can be obtained in the case of polytopal windows, that is when W is a convex polytope. The ideas of Julien where picked up by Koivusalo and Walton, [27], who where proved the following theorem. We will assume the stabilizers of the hyperplanes which bound the window are trivial; in the original theorem the role of this stabilizers is addressed.…”

“…Theorem 1.3 (Koivusalo, Walton, [27,Theorem 7.1]) Consider a model set Λ(R n , R d , Γ, W ) with a polytopal window W . Assume that the stabilizer of the hyperplanes which bound the window are trivial.…”