U. Grimm: Diffraction of a Pinwheel Tiling — Problem

Definitions and mathematical statement

Shared notation.

$\mathbb N=\{1,2,\ldots\}$, and $\mathbb Z,\mathbb Q,\mathbb R,\mathbb C$ denote the integers, rationals, reals and complex numbers. Any other conventions are specified locally.

A pinwheel tiling is the Conway--Radin substitution tiling by right triangles of side lengths $1,2,\sqrt5$: inflating by $\sqrt5$ yields five congruent original triangles in the standard pinwheel subdivision. Choose on each triangle the point whose coordinates from the right-angle vertex along its perpendicular unit axes are $(1/2,1/2)$, and let $\Lambda$ be all these points. The uniform Dirac comb is $\omega=\sum_{x\in\Lambda}\delta_x$. Its volume-averaged autocorrelation is \[\gamma=\lim_{R\to\infty}\frac{\omega|_{B_R}*\widetilde{\omega|_{B_R}}}{\operatorname{area}(B_R)},\] in the vague topology; the tilde reflects and conjugates a measure. The positive diffraction measure is its Fourier transform $\widehat\gamma$, with kernel $e^{-2\pi i k\cdot x}$. This measure is rotationally invariant. A sharp ring at radius $r>0$ is a nonzero component $I_r\sigma_r$ of $\widehat\gamma$, where $\sigma_r$ is normalized uniform measure on the circle $\{|k|=r\}$. Its intensity $I_r$ is its total measure; such a ring is an atom of the radial measure even though it is singular continuous as a measure on the plane.

Statement recorded in the supplied source.

Determine all sharp-ring radii of pinwheel diffraction and their intensities. [S2]

Short English statement

Locate the sharp circular diffraction rings and calculate how much intensity each carries.

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, UnsolvedMath: A Curated Collection of Open Mathematics Problems, record 6600008 (AMR-065-0008). CC BY 4.0. https://huggingface.co/datasets/ulamai/UnsolvedMath.
• S2 Faustin Adiceam (compiler), Open Problems and Conjectures related to the Theory of Mathematical Quasicrystals, arXiv:1604.06280v2 (2016), Section 2.3, construction and Problem 2.3.1, pp. 5--6. https://arxiv.org/pdf/1604.06280.