Let $H: [0,1] \to [0,1]\times[0,1]$
be the Hilbert curve (the limit of the family of functions defined on this page )
This curve is well known for its "good locality propreties"
Meaning that for $(x,y) \in [0,1]\times[0,1]$, If x and y are close, the distance between $H(x)$ and $H(y)$ will be small.
But I would need a more precise measure of how "local" this curve is.
If I'm not mistaken, the answer to this post indicates that
$$\forall x,y \in [0,1], ||H(x)-H(y)|| \leq 4\sqrt{|x-y|}$$
But experimentally, it seems that we can replace the value of 4 by around 2.4 (see this paper)
I'm surprised that nobody proved the value of a smallest constant $K$ such that
$$\forall x,y \in [0,1], ||H(x)-H(y)|| \leq K\sqrt{|x-y|}$$
Is there such a proof ? If not, is there a "reason" it is so complicated ?