Abstract
We calculate the growth rate of the complexity function for polytopal cut and project sets. This generalizes 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 by 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.
Highlights
Periodic patterns of Euclidean space have been central geometric objects of study in mathematics for millennia
This repairs the error in the main argument of [Jul10] in determining the growth of the complexity function for a polytopal cut and project set under the quasicanonical condition
Our work demonstrates that in such cases the almost canonical condition should be replaced with the quasicanonical condition
Summary
Periodic patterns of Euclidean space have been central geometric objects of study in mathematics for millennia. This repairs the error in the main argument of [Jul10] in determining the growth of the complexity function for a polytopal cut and project set under the quasicanonical condition To this end we prove the following result. We contrast the quasicanonical and almost canonical conditions through some simple examples, and demonstrate that in particular cases the connection between acceptance domains and cut regions fails to hold true if only assuming the almost canonical property We believe that these first sections could be of independent importance in the study of patterns in cut and project sets. F g: for two functions as above, f g means that f g and g f
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