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Planar Bilipschitz Extension from Separated Nets

Michael Dymond, Vojtěch Kaluža

TL;DR

This work addresses planar bilipschitz extension from discrete sets, proving that any $L$-bilipschitz map on $\mathbb{Z}^{2}$ (and more generally any separated net in $\mathbb{R}^{2}$) extends to a bilipschitz map on $\mathbb{R}^{2}$ with a polynomial bound on the extension constant. The authors develop a multi-stage strategy: approximate a shoreline on horizontal lines, separate discrete image sets with a bilipschitz curve, extend along wide horizontal strips, and glue the pieces using a Kovalev-style extension inside strips. The approach yields explicit polynomial bounds and resolves the two-dimensional Oberwolfach/open problems of Navas and of Alestalo–Trotsenko–Väisälä, while also outlining a path towards higher dimensions via a companion work. The main contribution is a complete, quantitative solution in dimension two, establishing that bilipschitz extensions from separated nets to the plane are always possible with controlled constants, and it links discrete-net extendability to strip-wise, geometrically controlled constructions.

Abstract

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.

Planar Bilipschitz Extension from Separated Nets

TL;DR

This work addresses planar bilipschitz extension from discrete sets, proving that any -bilipschitz map on (and more generally any separated net in ) extends to a bilipschitz map on with a polynomial bound on the extension constant. The authors develop a multi-stage strategy: approximate a shoreline on horizontal lines, separate discrete image sets with a bilipschitz curve, extend along wide horizontal strips, and glue the pieces using a Kovalev-style extension inside strips. The approach yields explicit polynomial bounds and resolves the two-dimensional Oberwolfach/open problems of Navas and of Alestalo–Trotsenko–Väisälä, while also outlining a path towards higher dimensions via a companion work. The main contribution is a complete, quantitative solution in dimension two, establishing that bilipschitz extensions from separated nets to the plane are always possible with controlled constants, and it links discrete-net extendability to strip-wise, geometrically controlled constructions.

Abstract

We prove that every -bilipschitz mapping can be extended to a -bilipschitz mapping and provide a polynomial upper bound for . Moreover, we extend the result to every separated net in instead of , 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.

Paper Structure

This paper contains 9 sections, 27 theorems, 120 equations, 2 figures.

Table of Contents

  1. Introduction.
  2. Acknowledgements.
  3. Preliminaries.
  4. Extensions.
  5. Gluing of maps.
  6. First approximation of the shore.
  7. Separation of discrete sets by a bilipschitz curve.
  8. Extension to horizontal lines.
  9. Bilipschitz mappings on a strip.

Key Result

Theorem 1.1

There is a polynomial function $p\colon \mathbb{R}\to \mathbb{R}$ such that for any $L\geq 1$ and any $L$-bilipschitz mapping $f\colon\mathbb{Z}^{2}\to\mathbb{R}^{2}$ there is a $p(L)$-bilipschitz mapping $F\colon\mathbb{R}^{2}\to\mathbb{R}^{2}$ such that $F|_{\mathbb{Z}^{2}}=f$.

Figures (2)

  • Figure 1: Definition of $\psi$.
  • Figure 2: The position of $Q_x$ with respect to $\phi(\mathbb{R})$.

Theorems & Definitions (62)

  • Theorem 1.1
  • Corollary 1
  • Theorem 1.2
  • Theorem 1.3
  • Theorem 1.3
  • Proposition 1
  • Theorem 1.3
  • Theorem 1.3
  • Theorem 1.4: Stronger version of Theorem \ref{['thm:mainres']}
  • proof : Proof of Theorems \ref{['thm:mainres']} and \ref{['thm:mainres2']}
  • ...and 52 more