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Source. Theorem 3, p. 15, 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 (p. 14). Let $\Lambda=\Lambda(W)$ be a model set (Section 2) with star map ${}^*:L\to G$, let $\Lambda_R:=\Lambda\cap B_R(0)$ and let $\mu$ be Haar measure on $G$. For a function $f^*:G\to\mathbb C$ define $f:L\to\mathbb C$ by $f(x)=f^*(x^*)$.

Theorem 3 (p. 15). Attributed to Weyl (reference [39]): if $\Lambda$ is regular and $f^*$ is continuous, then

$$\lim_{R\to\infty}\frac{1}{\mathrm{card}(\Lambda_R)}\sum_{x\in\Lambda_R}f(x) =\frac{1}{\mathrm{vol}(W)}\int_W f^*(u)\,d\mu(u)$$

(display (16)). The paper writes $\mathrm{vol}(W)$ without defining it separately; it is read here as the Haar measure $\mu(W)$.

The paper adds (p. 15) that, since $\partial W$ has measure zero, a function $f^*$ supported on $W$ need only be continuous on $W$ rather than on all of $G$.

Proof pointer

No proof is given in the survey. Section 5 (p. 14) introduces the passage from the model set to its window as H. Weyl's theory of uniform distribution, and states the theorem right after Theorem 2.

Dependencies

Theorem 2 and 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.