A characterization of linearly repetitive cut and project sets

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 London Mathematical Society* Research supported by EPSRC grants EP/L001462, EP/J00149X, EP/M023540. HK also gratefully acknowledges the support of the Osk. Huttunen foundation.

Statistics of patterns in typical cut and project sets

“…Lemma 2.1 (Lemma 2.4 of [12]). For a regular, cubical cut and project set, for every equivalence class P of r-patches there is a unique connected component Q of reg(r) such that, for any y ∈ Y , P(y, r) = P if and only if y * ∈ Q.…”

Open Problems and Conjectures Related to the Theory of Mathematical Quasicrystals

“…Let Y (α, β) be a canonical cut-and-project set (this concept is defined in Haynes et al 2016a) formed using the subspace…”

Perfectly ordered quasicrystals and the Littlewood conjecture

“…Perhaps more surprisingly, in contrast with Theorem 1.2, we obtain the existence of uncountably many 'super perfectly ordered' quasicrystals, when C d is replaced by R d . Our proof of this theorem also leads to an explicit method, described in [16,Section 6], for constructing such sets.…”

“…The proofs of our theorems are based on a collection of observations from tiling theory and Diophantine approximation, which have been developed in several recent works [6,15,16,17]. In [17] it was explained how one can translate the problem of studying patterns in cut and project sets to a dual problem of studying connected components of sets in the internal space, defined by a natural (linear) Z k -action.…”

“…physical and internal space pairs which give rise to Amman-Beenker and Penrose tilings), we are able to argue directly to understand the structure of canonical cut and project sets from the corresponding cubical ones. These issues are discussed in some detail in [16].…”