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Source. Theorem 9, p. 19, of Robert V. Moody, Model Sets: A Survey, in From Quasicrystals to More Complex Systems (Les Houches School lecture notes), Springer/EDP Sciences (2000), 145-166, doi:10.1007/978-3-662-04253-3_6, read in the preprint arXiv:math/0002020v1 (2 Feb 2000) named on the source card; pages here are that preprint's pages, and the book pagination was not compared.

Read depth. Claims checked: the statement and its setting were read clause by clause on the printed pages. The survey gives no proof. Nothing here is independently reviewed.

Statement

Setting (pp. 18-19). Let $\Lambda=\Lambda(W)=\{x\in L: x^*\in W\}$ be a model set (Section 2). Each $(u,v)\in\mathbb R^d\times G$ gives the model set $\Lambda(W,u,v):=u+\{x\in L: x^*\in -v+W\}$ (display (25)), and $(u,v)\in\tilde L$ gives back $\Lambda$, so $\mathbb T:=(\mathbb R^d\times G)/\tilde L$ parametrizes a family of model sets: the torus parametrization, a name taken from Baake, Hermisson and Pleasants (reference [3]), though $\mathbb T$ need not be a torus. $\mathbb R^d$ acts on $\mathbb T$ by $(x,y+\tilde L)\mapsto x+y+\tilde L$; its orbits correspond to model sets differing only by translation, and the action of $G$ moves the window.

Theorem 9 (p. 19). Let $\Lambda$ be a generic model set. Then the action $\mathbb R^d\times\mathbb T\to\mathbb T$ (display (26)) is a minimal, uniquely ergodic dynamical system $\mathcal D_{\rm tor}$, whose unique invariant probability measure is normalized Haar measure. The points of $\mathcal D_{\rm tor}$ corresponding to generic model sets form a dense set, and the points corresponding to non-generic model sets form a set of the first category.

The paper notes (p. 19) that $\mathcal D_{\rm tor}$ does not depend on $W$, though the parametrization of model sets by it does.

Proof pointer

No proof is given in the survey; Section 6 (pp. 16-17) states that its results may be found in Schlottmann's paper (reference [36]).

Dependencies

The definitions of Section 2.

Bears on

• Problem 188: the paper does not mention the problem. It is a survey of the cut-and-project construction of aperiodic point sets; it says nothing about unit distances, two-colorings of the plane or arithmetic progressions, and gives no coloring and no bound on the number of terms the problem asks about.