April 2023 EFFICIENTLY FILLING SPACE
Paul D. Humke, Khang V. Huynh, Thong H. Vo
Rocky Mountain J. Math. 53(2): 477-484 (April 2023). DOI: 10.1216/rmj.2023.53.477

Abstract

We show that for each n=3,4, there is a space-filling curve f:[0,1][0,1]n such that f is at most (n+1)-to-1 at every point of [0,1]n. The fact that any such dimension raising continuous function is at least (n+1)-to-1 has been known since the 1930s, so the examples we provide here are, in that sense, the best possible. The classic space-filling curves due to first Peano and a year later, Hilbert, that map [0,1] onto [0,1]2 are both 4-to-1 at a dense set of points and their generalizations to [0,1]n are known to be 2n-to-1 at a dense set of points. Flaten, Humke, Olson and Vo (J. Math. Anal. Appl. 500:2 (2021), art. id. 125113) gave an example, f:[0,1][0,1]2 based on the Hilbert linear ordering of somewhat altered Hilbert partitions which is at most 3-to-1 at every point of [0,1]2, but there are technical difficulties with generalizing that example to higher dimensions. In a sense, this paper represents an overcoming of those difficulties.

Citation

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Paul D. Humke. Khang V. Huynh. Thong H. Vo. "EFFICIENTLY FILLING SPACE." Rocky Mountain J. Math. 53 (2) 477 - 484, April 2023. https://doi.org/10.1216/rmj.2023.53.477

Information

Received: 18 April 2022; Revised: 21 June 2022; Accepted: 24 June 2022; Published: April 2023
First available in Project Euclid: 20 June 2023

MathSciNet: MR4604768
zbMATH: 07725149
Digital Object Identifier: 10.1216/rmj.2023.53.477

Subjects:
Primary: 26A03

Keywords: dimension raising functions , Lebesgue covering theorem , Peano curve , space-filling

Rights: Copyright © 2023 Rocky Mountain Mathematics Consortium

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Vol.53 • No. 2 • April 2023
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