Abstract
For the development of a mathematical theory which can be used to rigorously investigate physical properties of quasicrystals, it is necessary to understand regularity of patterns in special classes of aperiodic point sets in Euclidean space. In one dimension, prototypical mathematical models for quasicrystals are provided by Sturmian sequences and by point sets generated by substitution rules. Regularity properties of such sets are well understood, thanks mostly to well known results by Morse and Hedlund, and physicists have used this understanding to study one dimensional random Schrödinger operators and lattice gas models. A key fact which plays an important role in these problems is the existence of a subadditive ergodic theorem, which is guaranteed when the corresponding point set is linearly repetitive.In this paper we extend the one-dimensional model to cut and project sets, which generalize Sturmian sequences in higher dimensions, and which are frequently used in mathematical and physical literature as models for higher dimensional quasicrystals. By using a combination of algebraic, geometric, and dynamical techniques, together with input from higher dimensional Diophantine approximation, we give a complete characterization of all linearly repetitive cut and project sets with cubical windows. We also prove that these are precisely the collection of such sets which satisfy subadditive ergodic theorems. The results are explicit enough to allow us to apply them to known classical models, and to construct linearly repetitive cut and project sets in all pairs of dimensions and codimensions in which they exist.
Highlights
A Delone set Y ⊆ Rd is linearly repetitive (LR) if there exists a constant C > 0 such that, for any r ≥ 1, every patch of size r in Y occurs in every ball of diameter Cr in Rd
The characterization which we provide in Theorem 1.1 opens the door for the study of physical properties of a rich collection of higher dimensional point sets, a project which is the focus of our current research on this topic
In [25, Problem 8.3], motivated by many of the connections which we have described, Lagarias and Pleasants asked for a characterization of LR cut and project sets
Summary
A Delone set Y ⊆ Rd is linearly repetitive (LR) if there exists a constant C > 0 such that, for any r ≥ 1, every patch of size r in Y occurs in every ball of diameter Cr in Rd. It should be pointed out that most of the canonical cut and project sets of specific interest in the literature arise from subspaces defined by linear forms with coefficients in a fixed algebraic number field In such a case the Diophantine behavior alluded to at the end of the paragraph before Theorem 1.2 cannot occur. Note that in the special case when k −d = 1, condition (LR1) is automatically satisfied, and condition (LR2) requires the linear form defining Y to be badly approximable in the usual sense This observation, together with Proposition 4.2, immediately implies the following complete characterization of both cubical and canonical cut and project sets in codimension one. A k to k − 1 cubical or canonical cut and project set defined by a linear form L is linearly repetitive if and only if L is badly approximable
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