Cut and project sets with polytopal window II: linear repetitivity

TL;DR

This paper classifies when aperiodic cut and project sets with convex polytopal windows exhibit linear repetitivity, linking it to low complexity patterns and a Diophantine condition on the projected lattice.

Contribution

It provides a complete classification of linear repetitivity for a broad class of polytopal cut and project schemes, extending previous cubical scheme results.

Findings

LR holds if and only if patterns have low complexity and a Diophantine condition is satisfied.

The paper develops new decomposition techniques for polytopal schemes and relates them to Diophantine approximation.

Examples demonstrate the main theorem and its implications for ergodic properties.

Abstract

This paper gives a complete classification of linear repetitivity (LR) for a natural class of aperiodic Euclidean cut and project schemes with convex polytopal windows. Our results cover those cut and project schemes for which the lattice projects densely into the internal space and (possibly after translation) hits each supporting hyperplane of the polytopal window. Our main result is that LR is satisfied if and only if the patterns are of low complexity (property C), and the projected lattice satisfies a Diophantine condition (property D). Property C can be checked by computation of the ranks and dimensions of linear spans of the stabiliser subgroups of the supporting hyperplanes, as investigated in Part I to this article. To define the correct Diophantine condition D, we establish new results on decomposing polytopal cut and project schemes to factors, developing concepts initiated in the work of Forrest, Hunton and Kellendonk. This means that, when C is satisfied, the window splits into components which induce a compatible splitting of the lattice. Then property D is the requirement that, for any suitable decomposition, these factors do not project close to the origin in the internal space, relative to the norm in the total space. On each factor, this corresponds to the usual notion from Diophantine Approximation of a system of linear forms being badly approximable. This extends previous work on cubical cut and project schemes to a very general class of cut and project schemes. We demonstrate our main theorem on several examples, and derive some further consequences of our main theorem, such as the equivalence LR, positivity of weights and satisfying a subadditive ergodic theorem for this class of polytopal cut and project sets.