Year: 2026
Type: preprint
Abstract: Following a hint of Semmes, we employ Stong's bijections between integer lattices to construct space-filling surfaces, which are higher-dimensional analogues of space-filling curves. For each $m\geq 2$ we build $α$-Hölder continuous parameterizations $f:[0,1]^m\rightarrow[0,1]^{m+1}$ with sharp exponent $α=m/(m+1)$. In particular, there exist $(2/3)$-Hölder continuous surjections from squares to cubes. This solves Arnold's problem 1988--5.
Source: arXiv (Cornell University)
Authors: Badger, Matthew, Palmer, Kevin
Subfield: Mathematical Physics
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W7204123997
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