Additive Euclidean spanner on the integer lattice

Determine whether there exists an unweighted graph H with vertex set Z^2 such that the supremum over all pairs u,v in Z^2 of d_H(u,v) minus their Euclidean distance d(u,v) is finite.

Background

The paper studies graphs whose shortest-path distances approximate Euclidean distances between vertices. Its constructions establish asymptotically vanishing relative error, including bounds of the form d_H(u,v)=d(u,v)+O(d(u,v)/log{2/3} d(u,v)) and polynomial additive error on enlarged grids, but they do not establish a uniformly bounded additive error on the standard integer lattice.

The unresolved question asks whether the Euclidean plane, represented by the vertex set Z2, admits an unweighted graph metric whose excess over Euclidean distance is bounded by one absolute constant for every pair of lattice points. The displayed problem requires precisely that the supremum of d_H(u,v)-d(u,v) over all lattice-point pairs be finite.

References

Motivated by the above considerations, several researchers, including Bruce Kleiner, Gady Kozma, Oded Schramm, Itai Benjamini, Paul Erdős and the first named author, independently, raised the following question, which is still open.

Does there exist an unweighted graph H on the vertex set V(H)=\mathbb{Z}2 such that \sup_{u,v\in \mathbb{Z}2}\left(d_H(u,v)-d(u,v)\right)<\infty\;?

— Cutting a convex body into fat parts and approximating Euclidean distance by graph distances  (2609.20702 - Pach et al., 17 Sep 2026) in Problem 1 (labelled "additive"), Historical remarks, background