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Extending Bilipschitz Mappings between Separated Nets

Michael Dymond, Vojtěch Kaluža

TL;DR

The paper develops a comprehensive framework to extend bilipschitz mappings from separated nets in ${\mathbb R}^d$ to the whole space, clarifying the longstanding open problem and achieving a complete positive solution in dimension ${2}$ via a companion work. It reduces the general extension problem to the lattice case ${\mathbb Z}^d$ through a sequence of controlled bilipschitz changes of variables and introduces a robust toolkit—bilipschitz swapping, gluing, and hyperplane threading—to manage extensions. Central contributions include a precise equivalence between lattice and general-net extendability, a detailed threading construction that preserves topological separations, and a reduction strategy that ties the extension problem to two conjectures about hyperplane-based extendability. The results lay groundwork for higher-dimensional progress by isolating core geometric and combinatorial obstacles and providing explicit bilipschitz bounds that guide future constructions and proofs.

Abstract

We provide a new characterisation of the decades old open problem of extending bilipschitz mappings given on a Euclidean separated net. In particular, this allows for the complete positive solution of the open problem in dimension two. Along the way, we develop a set of tools for bilipschitz extensions of mappings between subsets of Euclidean spaces.

Extending Bilipschitz Mappings between Separated Nets

TL;DR

The paper develops a comprehensive framework to extend bilipschitz mappings from separated nets in to the whole space, clarifying the longstanding open problem and achieving a complete positive solution in dimension via a companion work. It reduces the general extension problem to the lattice case through a sequence of controlled bilipschitz changes of variables and introduces a robust toolkit—bilipschitz swapping, gluing, and hyperplane threading—to manage extensions. Central contributions include a precise equivalence between lattice and general-net extendability, a detailed threading construction that preserves topological separations, and a reduction strategy that ties the extension problem to two conjectures about hyperplane-based extendability. The results lay groundwork for higher-dimensional progress by isolating core geometric and combinatorial obstacles and providing explicit bilipschitz bounds that guide future constructions and proofs.

Abstract

We provide a new characterisation of the decades old open problem of extending bilipschitz mappings given on a Euclidean separated net. In particular, this allows for the complete positive solution of the open problem in dimension two. Along the way, we develop a set of tools for bilipschitz extensions of mappings between subsets of Euclidean spaces.

Paper Structure

This paper contains 11 sections, 18 theorems, 76 equations.

Table of Contents

  1. Introduction
  2. Notation
  3. Sets in $\mathbb{R}^d$.
  4. Mappings.
  5. Bilipschitz Swapping and Gluing.
  6. Reduction to the integer lattice.
  7. Threading a bilipschitz cord through the channel.
  8. Construction of $\Xi$.
  9. Construction of $\Omega$.
  10. Properties of $\Psi:=\Omega\circ \Xi$.
  11. Two conjectures.

Key Result

Theorem 1.1

Let $d\in \mathbb{N}$. Then the following are equivalent: Furthermore, if Zd holds with $\operatorname{bilip}(F)\leq C_{d}\left(\operatorname{bilip}(f)\right)$ for some monotone increasing function $C_{d}\colon [1,\infty)\to [1,\infty)$ then gen_sep_net holds with where $K:= 16\max\left\{\frac{3d}{r},1\right\}$ and $R, r$ denote the net and separation constants of $A$ respectively.

Theorems & Definitions (37)

  • Theorem 1.1
  • Theorem 1.2
  • Conjecture 1
  • Conjecture 2
  • Theorem 1.3
  • Lemma 2
  • Lemma 2
  • proof : Proof of Theorem \ref{['thm:Zd_equiv']} (Sketch)
  • Theorem 1.3
  • proof
  • ...and 27 more