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. Put $\|a\|=\min_{m\in\mathbb Z}|a-m|$. The Littlewood condition for $(\alpha,\beta)$ is $\liminf_{n\to\infty}n\|n\alpha\|\|n\beta\|=0$.

Statement recorded in the supplied source.

For every $(\alpha,\beta)$ with $1,\alpha,\beta$ linearly independent over $\mathbb Q$, does the Littlewood condition imply $\sup_{\Omega\in\mathcal R}\#\xi_{(\alpha,\beta)}(\Omega)=\infty$? [S2]

Short English statement

Does Littlewood’s approximation condition force unbounded numbers of patch-frequency values over aligned rectangles?

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, UnsolvedMath: A Curated Collection of Open Mathematics Problems, record 6600012 (AMR-065-0012). 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 2.4.3, p. 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.