EFFICIENTLY FILLING SPACE

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.