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Source. The unnumbered definitions of Section 2 (pp. 4-6) and the definition of a Meyer set in Section 3 (p. 7) 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 definitions were read clause by clause on the printed pages. Nothing here is independently reviewed.

Statement

Cut and project scheme (p. 4, display (1)). It consists of a real Euclidean space $\mathbb R^d$ (the physical space), a locally compact abelian group $G$ (the internal space), the projections $\pi_1:\mathbb R^d\times G\to\mathbb R^d$ and $\pi_2:\mathbb R^d\times G\to G$, and a lattice $\tilde L\subset\mathbb R^d\times G$, that is, a discrete subgroup with $(\mathbb R^d\times G)/\tilde L$ compact. It is assumed that $\pi_1$ restricted to $\tilde L$ is injective and that $\pi_2(\tilde L)$ is dense in $G$. With $L=\pi_1(\tilde L)$, the star map ${}^*:L\to G$ sends $x$ to $\pi_2\big((\pi_1|_{\tilde L})^{-1}(x)\big)$ (display (2), where the print writes the inverse as $\pi_1|_L^{-1}$).

Model set (pp. 4-5, display (3) and condition W1). For $W\subset G$, $\Lambda(W)=\{\pi_1(x): x\in\tilde L,\ \pi_2(x)\in W\}=\{u\in L: u^*\in W\}$. Such a set, or any translate of it, is a model set (or cut and project set) when the window satisfies

• W1 (p. 5): $W$ is nonempty and $W=\overline{\operatorname{int}(W)}$ is compact.

The paper remarks that the equality in W1 could be replaced by an inclusion, and that it keeps the equality because then $\overline{\Lambda^*}=W$ (p. 5).

Two further conditions are used for the deeper results (p. 5):

• W2: the model set is generic if the boundary of its window meets $\pi_2(\tilde L)$ in no point, $\partial W\cap\pi_2(\tilde L)=\emptyset$.
• W3: the model set is regular if $\partial W$ has Haar measure $0$.

A footnote (p. 5) warns that this terminology is not standard: what the paper calls generic is sometimes called regular elsewhere.

Torus (p. 6). $\mathbb T:=(\mathbb R^d\times G)/\tilde L$, a compact abelian group on which $\mathbb R^d$ acts; it is a torus when $G$ is a real space. The dual picture (display (5)) has the dual group $\hat{\mathbb T}$ as a lattice in $\widehat{\mathbb R^d}\times\hat G$, with canonical projections $\hat\pi_1$ and $\hat\pi_2$.

Meyer set (pp. 6-7). Model sets are Delone sets with finite local complexity, and in fact $\Lambda-\Lambda$ is uniformly discrete (display (6)); a Delone set with this property is a Meyer set. A Delone set is one that is uniformly discrete and relatively dense (p. 2).

Proof pointer

These are definitions. The geometric facts stated with them (Delone, finite local complexity, display (6)) are recorded on pp. 6-7 with references to other papers and are not proved in the survey.

Dependencies

None within the paper.

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.