Additive-error graph model of the Euclidean plane

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 $G=(V,E)$ be a connected graph whose edges have unit length, and let $d_G$ be shortest-path distance. A map $f:V\to\mathbb R^2$ is an additive-error rough isometry if some $C<\infty$ satisfies \[\left|d_G(x,y)-\|f(x)-f(y)\|_2\right|<C\quad(x,y\in V),\] and every point of $\mathbb R^2$ lies at Euclidean distance less than $C$ from $f(V)$. A bounded-degree graph has $\sup_{v\in V}\deg(v)<\infty$.

Statement recorded in the supplied source.

Does such a graph and additive-error rough isometry exist? Can $G$ in such an example have bounded degree? [S2]

Short English statement

Can the whole Euclidean plane be represented by a unit-edge graph with uniformly bounded additive distance error, including with bounded degrees?

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, UnsolvedMath: A Curated Collection of Open Mathematics Problems, record 10000005 (AMR-099-0005). CC BY 4.0. https://huggingface.co/datasets/ulamai/UnsolvedMath.
• S2 Itai Benjamini, Coarse Geometry and Randomness, Saint-Flour lecture notes, version dated 30 October 2013; original author manuscript preserved in Arquivo.pt capture of 31 December 2020, Definition 1.25 and Open Problem 1.32, pp. 9--10. https://arquivo.pt/noFrame/replay/20201231041548id_/http://www.wisdom.weizmann.ac.il/~itai/stflouraug24.pdf.