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Source. Theorem 12, p. 21, 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, its setting and the proof sketch were read on the printed pages. Nothing here is independently reviewed.

Statement

Setting (pp. 20-21). For a regular model set $\Lambda$ (Section 2), let $\delta_\Lambda=\sum_{x\in\Lambda}\delta_x$ and $\Lambda_s=\Lambda\cap B_s(0)$. The autocorrelation measure is the vague limit $\gamma=\lim_{s\to\infty}\frac{1}{\mathrm{vol}(B_s(0))}\sum_{x,y\in\Lambda_s}\delta_{x-y}$ (display (30)); its Fourier transform $\hat\gamma$ is a positive measure, the diffraction pattern. Its point part is the Bragg spectrum, and $\Lambda$ has pure point spectrum when the continuous part is $0$. The dual group $\hat{\mathbb T}$ and the projection $\hat\pi_1$ are those of the dual picture (display (5), p. 6).

Theorem 12 (p. 21, quoted). "Any regular model set has pure point spectrum. Furthermore this spectrum is supported on the projection into Fourier space on the physical side of the dual of the compact group $\mathbb T$ (5), i.e. it has the form"

$$\hat\gamma=\sum_{k\in\hat{\mathbb T}}w(k)\,\delta_{\hat\pi_1(k)}$$

(display (31)). The survey attributes the theorem to Schlottmann (reference [36]); the weights $w(k)$ are given by Theorem 13.

Proof pointer

Pages 21-22, following an idea of Dworkin as written out by Hof. One may take $\Lambda$ generic, since translating the window does not change the qualitative nature of the diffraction. Smoothing $\delta_\Lambda$ by a bump function $b$ gives a continuous function $\psi$ on the hull $\mathcal D(\Lambda)$; the Birkhoff ergodic theorem, with unique ergodicity, turns the autocorrelation of $b*\delta_\Lambda$ into the correlation $(T_x\psi,\psi)$ on $\mathcal D(\Lambda)$, and Theorem 11 (which transfers the discrete spectrum of $\mathcal D_{\rm tor}$ to $\mathcal D(\Lambda)$) makes its Fourier transform pure point. Letting $b$ tend to $\delta_0$ gives the theorem.

Dependencies

Theorems 8, 10 and 11 of the survey (Section 6, pp. 18-20), Theorem 9, 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.