A. Navas: A Conjecture on Delone Sets BL to Lattices (after P. Alestalo, D.A. Trotsenko and J. V\"ais\"al\"a). — 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 Delone set $\mathcal D\subset\mathbb R^2$ is uniformly discrete (distinct points are separated by some $r>0$) and relatively dense (every point of the plane lies within a fixed finite distance of $\mathcal D$). It is bi-Lipschitz equivalent (BL) to $\mathbb Z^2$ if there is a bijection $b:\mathcal D\to\mathbb Z^2$ and $C\geq1$ such that $C^{-1}|x-y|\leq|b(x)-b(y)|\leq C|x-y|$ for all $x,y\in\mathcal D$. A bi-Lipschitz ambient map satisfies the same two-sided bound for all points of the plane.

Statement recorded in the supplied source.

If $\mathcal D$ is BL to $\mathbb Z^2$, must there exist a bi-Lipschitz map $L:\mathbb R^2\to\mathbb R^2$ such that $L(\mathcal D)=\mathbb Z^2$? [S2]

Short English statement

Can a bi-Lipschitz correspondence between a planar Delone set and the lattice be realized by a bi-Lipschitz map of the whole plane?

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, UnsolvedMath: A Curated Collection of Open Mathematics Problems, record 6600014 (AMR-065-0014). 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 2.6, Problem 2.6.1, pp. 8--9. https://arxiv.org/pdf/1604.06280.