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Source. Theorem 13, p. 22, 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. 20-22). $\Lambda=\Lambda(W)$ is a regular model set (Section 7 fixes this on p. 20), $\hat{\mathbb T}$ is the dual group of $\mathbb T=(\mathbb R^d\times G)/\tilde L$, $\hat\pi_2$ is the projection to $\hat G$ in the dual picture (display (5), p. 6), and $w(k)$ is the weight of the Bragg peak at $\hat\pi_1(k)$ in Theorem 12.

Theorem 13 (p. 22, quoted). "Let $k\in\hat{\mathbb T}$ and let $\chi$ denote the characteristic (or indicator) function of $W$. Then $w(k)=|\hat\chi(-\hat\pi_2(k))/\mathrm{vol}(W)|^2$."

The survey calls this the quantitative counterpart of Theorem 12 and attributes it to Meyer (reference [26]).

Proof pointer

No proof is given in the survey.

Dependencies

Theorem 12 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.