Is there a space-filling curve $C$ that has the property that, if $C$ passes through $p_1=(x_1,y_1)$ at a distance $d_1$ along the curve, and through $p_2$ at $d_2$, then if $|p_1 - p_2| \le a$, then $|d_1 - d_2| \le b$, for some constants $a$ and $b$? In other words, any two points of the plane within distance $a$ are separated by at most $b$ along $C$. Call this property distance locality. So I am asking whether a curve exists mapping $\mathbb{R}$ to $\mathbb{R}^2$ with distance locality.
Although I doubt the answer differs, permit me also to ask the same question for $\mathbb{Q}^2$, and for $\mathbb{Z}^2$.
I have little experience with the properties of the known space-filling curves. Those better schooled on this topic can likely answer these questions easily. Thanks!
Addendum. I noticed a paper just released today which is focused on "locality properties" of 3D space-filling curves: "An inventory of three-dimensional Hilbert space-filling curves," arXiv:1109.2323v1 [cs.CG]. The author explores several different locality measures that have been considered in the literature, and cites a wealth of references.