# Cut and project sets with polytopal window I: complexity

## TL;DR

This paper investigates the complexity growth of polytopal cut and project sets, correcting previous misconceptions and introducing a new quasicanonical condition that influences their topological analysis.

## Contribution

It introduces the quasicanonical condition, clarifies the relationship between acceptance domains and cut regions, and generalizes complexity calculations beyond the almost canonical case.

## Key findings

- Corrects errors in the literature regarding acceptance domains and cut regions.
- Introduces the quasicanonical condition ensuring proper relation between domains.
- Provides complexity growth rates for a broader class of polytopal cut and project sets.

## 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.