A. Haynes: Gaps Problems — 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.

Let $E=\{(x,y,\alpha x+\beta y):x,y\in\mathbb R\}\subset\mathbb R^3$, with $1,\alpha,\beta$ linearly independent over $\mathbb Q$. Write $F=\{(0,0,z):z\in\mathbb R\}$ and let $\rho,\rho^*$ be the complementary projections onto $E,F$. The canonical window is $W=\rho^*([0,1]^3)$ and the strip is $S=E+W$. Choose a physical projection $\pi:\mathbb R^3\to E$ whose kernel complements $E$ and whose restriction to $\mathbb Z^3$ is injective. For a regular shift $s$ (no point of $\mathbb Z^3+s$ on $\partial S$), put $Y=\pi(S\cap(\mathbb Z^3+s))$. The canonical system is aperiodic and totally irrational. For $y\in Y$, denote its unique lattice lift by $\widetilde y\in S\cap(\mathbb Z^3+s)$. A type-2 patch of shape $\Omega\subset E$ anchored at $y$ is \[P_2(y,\Omega)=\{y'\in Y:\rho(\widetilde y'-\widetilde y)\in\Omega\}.\] Two anchored patches have the same type when $P_2(y,\Omega)-y=P_2(z,\Omega)-z$. Its frequency is the limiting proportion, among anchors $z\in Y\cap B_R$, having that type as $R\to\infty$. Let $\xi_{(\alpha,\beta)}(\Omega)$ be the set of distinct such frequency values, rather than the set of patch types. Aligned rectangles are images in $E$ of coordinate rectangles in the $(x,y)$ plane; aligned squares have equal coordinate side lengths. Thus alignment and squareness use these graph coordinates. These conventions are given in the source’s cited definitions.

Statement recorded in the supplied source.

Does there exist $(\alpha,\beta)$ with $1,\alpha,\beta$ linearly independent over $\mathbb Q$ for which $\sup_{\Omega\in\mathcal S}\#\xi_{(\alpha,\beta)}(\Omega)=\infty$, where $\mathcal S$ consists of all aligned squares? [S2]

Short English statement

Can one canonical cut-and-project set have arbitrarily many different patch-frequency values as the aligned square shape varies?

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, UnsolvedMath: A Curated Collection of Open Mathematics Problems, record 6600010 (AMR-065-0010). 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 2.4, Problem 2.4.1, pp. 6--7. https://arxiv.org/pdf/1604.06280.
• S3 Alan Haynes, Henna Koivusalo, Lorenzo Sadun and James Walton, Gaps problems and frequencies of patches in cut and project sets, arXiv:1411.0578 (2014), Section 1.2, pp. 2--5. https://arxiv.org/pdf/1411.0578.
• S4 Alan Haynes, Henna Koivusalo and James Walton, Perfectly ordered quasicrystals and the Littlewood conjecture, arXiv:1506.05649v3 (2015), Sections 1.1 and 2.1, pp. 1--2 and 6--8. https://arxiv.org/pdf/1506.05649.