B. Weiss — 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 $1\leq d<k$ and let $E\subset\mathbb R^k$ be a $d$-plane which is totally irrational, meaning its image is dense in $\mathbb R^k/\mathbb Z^k$. Choose a complementary internal subspace $F$ and projections $\pi_E,\pi_F$ along this decomposition, with $\pi_E|_{\mathbb Z^k}$ injective. A cut-and-project set from a window $W\subset F$ and a shift $s$ is \[Y=\{\pi_E(z):z\in\mathbb Z^k+s,\ \pi_F(z)\in W\}.\] Here $W$ is bounded, has nonempty interior, and its boundary has $(k-d)$-dimensional Lebesgue measure zero. A full-rank lattice in $E$ is generated over $\mathbb Z$ by a basis of $E$. BL equivalence means a bijection $b:Y\to L$ with $C^{-1}|y-y'|\leq|b(y)-b(y')|\leq C|y-y'|$ for a finite $C\geq1$.

Statement recorded in the supplied source.

Is every cut-and-project set $Y$ under these hypotheses BL equivalent to a lattice in $E$? [S2]

Short English statement

Must every totally irrational cut-and-project set with this regular bounded window have a bi-Lipschitz correspondence to a lattice?

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, UnsolvedMath: A Curated Collection of Open Mathematics Problems, record 6600020 (AMR-065-0020). 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 2.8.2, Problem 2.8.2, p. 11. https://arxiv.org/pdf/1604.06280.
• S3 Alan Haynes, Henna Koivusalo and James Walton, Perfectly ordered quasicrystals and the Littlewood conjecture, arXiv:1506.05649v3 (2015), Section 2.1, pp. 6--8. https://arxiv.org/pdf/1506.05649.