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Source. Theorem 2, p. 14, 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 the definition it uses were read clause by clause on the printed pages. The survey gives no proof. Nothing here is independently reviewed.

Statement

Setting (p. 14). Let $\Lambda=\Lambda(W)$ be a model set (Section 2) and, for $R>0$, let $\Lambda_R:=\Lambda\cap B_R(0)$, where $B_R(0)$ is the ball of radius $R$ about the origin of $\mathbb R^d$. Let $\mu$ be Haar measure on $G$. The sets $\Lambda_R^*$ are called uniformly distributed when display (15) holds for each open set $U\subset W$; it is printed as "$\lim_{R\to\infty}\frac{\mathrm{card}(\Lambda_R^*\cap U)}{\mu(W)} \;=\; \mu(U)/\mu(W)$" [sic]. Read literally, the left side has no finite limit once $\Lambda_R^*\cap U$ grows without bound; the intended denominator is evidently $\mathrm{card}(\Lambda_R^*)$, so that the proportion of the points of $\Lambda_R^*$ lying in $U$ tends to $\mu(U)/\mu(W)$ (a reading of this page, not of the paper).

Theorem 2 (p. 14, quoted). "If $\Lambda$ is regular then the sets $\Lambda_R^*$ are uniformly distributed over $W$."

Regular means that $\partial W$ has Haar measure $0$ (condition W3, p. 5). The survey attributes the theorem to Schlottmann and to Hof (its references [35] and [19]).

Proof pointer

No proof is given in the survey. Section 3 (p. 7) notes that the well-defined positive frequency of each finite patch of a regular model set is not hard to prove once this theorem is established.

Dependencies

The definitions of Section 2. Theorem 3 is its averaging form.

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.