I am trying to figure out if there is a nice formula for a space-filling curve.
Define a subelementary function as one composed of:
- addition, subtraction, multiplication,
- $\sin$,
- the absolute value function,
- variables,
- $\pi$,
- and constants for rational numbers.
Key to the proof of Richardson's theorem in the one-variable case is that there is a subelementary map $x \mapsto (x \sin(x), x \sin(x^3))$ that gets arbitrarily close to any point in $\mathbb{R}^2$.
Is it possible to improve this to a map $x \mapsto (f(x),g(x))$ with $f$ and $g$ subelementary that achieves every point in $\mathbb{R}^2$?
Reading the answers to Is it true that a space-filling curve cannot be injective everywhere?, it seems like such a function cannot be injective, since all subelementary functions must be continuous. But I'm only interested in a surjective function.
