Planar Bilipschitz Extension from Separated Nets

We prove that every $L$-bilipschitz mapping $\mathbb{Z}^2\to\mathbb{R}^2$ can be extended to a $C(L)$-bilipschitz mapping $\mathbb{R}^2\to\mathbb{R}^2$ and provide a polynomial upper bound for $C(L)$. Moreover, we extend the result to every separated net in $\mathbb{R}^2$ instead of $\mathbb{Z}^2$, with the upper bound gaining a polynomial dependence on the separation and net constants associated to the given separated net. This answers an Oberwolfach question of Navas from 2015 and is also a positive solution of the two-dimensional form of a decades old open (in all dimensions at least two) problem due to Alestalo, Trotsenko and Väisälä.

💡 Research Summary

The paper addresses the long‑standing problem of extending bilipschitz embeddings defined on discrete subsets of the plane to the whole plane, with quantitative control of the bilipschitz constant. The authors prove that any L‑bilipschitz map f : ℤ² → ℝ² can be extended to a global bilipschitz homeomorphism F : ℝ² → ℝ² whose bilipschitz constant is bounded by a polynomial p(L). An explicit bound is given in Theorem 1.5: bilip(F) ≤ 10²⁰⁰⁰⁰ · L⁶⁰⁰⁰. Moreover, the result is generalized to arbitrary separated nets (Delone sets) X ⊂ ℝ². For an r‑separated, R‑net X, any L‑bilipschitz map f : X → ℝ² admits an extension with bilip(F) ≤ K · p(L) · R² · K³ · L, where K = 16·max{6/r, 1}.

The proof strategy builds on a companion paper