EFFICIENTLY FILLING SPACE

Abstract

We show that for each $n=3,4,\dots$ there is a space-filling curve $f:[0,1]\to {[0,1]}^n$ 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. 500:2 (2021), art. id. 125113) gave an example, $f:[0,1]\to {[0,1]}^2$ 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.