Hurewicz Lower Bound
Prove that any surjective continuous f: [0,1] → [0,1]² must be at least (k+1)-to-1 at a dense set, using the Cantor manifold theorem. This establishes that our construction is optimal - no curve can do better.
Journal Article - Published
Efficiently Filling Space
Rocky Mountain Journal of Mathematics · Vol. 53, No. 2 (2023) · Real Analysis
* Equal contribution
St. Olaf College
Theorem 1 - Hurewicz (1933)
Lower Bound: (k+1)-to-1 is Unavoidable
\[\text{Any continuous surjection}\; f:[0,1]\to [0,1]^k\]
\[\text{is } (k{+}1)\text{-to-1 at a countably dense set}\]
Theorem 3 - Main Result
Upper Bound: (k+1)-to-1 is Achievable
\[\forall\, k \geq 2,\;\exists\; f:[0,1] \to [0,1]^k\]
\[\text{at most } (k{+}1)\text{-to-1 everywhere on } [0,1]^k\]
\[\text{exactly } (k{+}1)\text{-to-1 on a dense set}\]
Comparison
Beating the Classical Constructions
\[\begin{aligned}
\text{Peano / Hilbert} &: 2^k\text{-to-1 at a dense set} \\[6pt]
\text{This paper} &: (k{+}1)\text{-to-1 everywhere}
\end{aligned}\]
Overview
Optimal space-filling curves for every dimension
Space-filling curves - continuous surjections \(f:[0,1] \to [0,1]^k\) - are necessarily "many-to-one" at many points: Hurewicz proved in 1933 that any such curve must be at least \((k+1)\)-to-1 at a countably dense set of points in the range. Classic constructions by Peano and Hilbert, however, fall far short of this bound - their natural generalizations to \([0,1]^k\) are \(2^k\)-to-1 at a dense set.
A prior result (Flaten, Humke, Olson, Vo 2021) showed that the \((k+1)\)-to-1 bound is achievable in dimension 2 using modified Hilbert partitions, but extending that construction to higher dimensions faced technical obstructions. This paper overcomes those difficulties.
For every \(k \geq 2\), we construct a space-filling curve \(f:[0,1] \to [0,1]^k\) that is at most \((k+1)\)-to-1 at every point - exactly matching the Hurewicz lower bound. The construction uses Lebesgue partitions, whose defining property is that no point is covered by more than \(k+1\) bricks, combined with Hilbert's nested-partition framework.
Method
From Lebesgue partitions to an optimal space-filling curve
Lebesgue Partitions
Invoke the Lebesgue covering theorem: [0,1]^k can be partitioned into arbitrarily fine closed "bricks" such that no point belongs to more than k+1 bricks. Points in fewer bricks will be lower-multiplicity preimage points.
Disjoint Hyperplane Construction
Prove (Theorem 4) that infinitely many ordered, linearized Lebesgue partitions of [0,1]^k exist with pairwise disjoint hyperplane boundary sets. This inductive construction ensures no spurious high-multiplicity points arise when partitions are nested.
Proper Sequence & Limit
Assemble the Lebesgue partitions into a proper sequence satisfying the adjacency, nesting, and diameter conditions of Hilbert's framework. Apply Theorem 2 to extract a continuous surjection that inherits the at-most-(k+1)-to-1 guarantee from the Lebesgue partition structure.
Results
Key findings for optimal space-filling curves
Optimal Multiplicity
For every k ≥ 2, the constructed curve is at most (k+1)-to-1 everywhere on [0,1]^k, exactly matching the Hurewicz lower bound. The classic Peano and Hilbert generalizations are 2^k-to-1 at a dense set - exponentially worse than optimal for large k.
Rich Preimage Structure
The curve is 1-to-1 on a residual (topologically "generic") subset of [0,1], exactly (k+1)-to-1 on a countable dense set, and [0,1] is its unique space-filling core. The full multiplicity classification mirrors what is known for the planar case.
Universal Construction
The Lebesgue partition approach works uniformly for all dimensions k ≥ 2, overcoming the dimensional obstruction that prevented the 2D technique (Flaten, Humke, Olson, Vo) from generalizing. The inductive step in Theorem 4 is the key new contribution.
Resources
Project materials
Citation
BibTeX
@article{humke2023efficiently,
title = {Efficiently Filling Space},
author = {Humke, Paul D. and Huynh, Khang Vo and Vo, Thong H.},
journal = {Rocky Mountain Journal of Mathematics},
volume = {53},
number = {2},
pages = {477--484},
year = {2023}
}