Efficiently Filling Space

Paul Humke, Khang Vo Huynh, Thong Hien Vo

Abstract:

We show that for each $n = 3, 4, \dots$ there is a space-filling curve
$f \colon [0,1] \to [0,1]^n$ 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 $2^n$-to-1 at a dense set of
points. Flaten, Humke, Olson and Vo ($J.\ Math.\ Anal.\ Appl.\ \mathbf{500}:2$
(2021), art. id. 125113) gave an example,
$f \colon [0,1] \to [0,1]^2$ 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.