To my understanding, quantitative topology/geometry makes statements quantitative. Examples: 1. a quantitative version of Invariance of Dimension is waist inequality. 2. Lusternik-Fet says a closed manifold has closed geodesic. A quantitative version is Gromov's systolic inequality. I'm looking for analogous statements about space-filling curves/trees. For example, Peano curve fills in space in the limit, we can ask: how quickly does it fill in?
Let $I = [0, 1]^2$ be a unit square on the plane, and $\{ P_{i} \}_{i=1}^{N}$ denote Peano curve sequence up to step $N$. One way to say its filling speed is to consider its $\epsilon$ perturbation: $\{ P_{i} + \epsilon \}_{i=1}^{N}$, by considering an $\epsilon$ tube around the curve. Now we can ask its area growth by looking at the following area ratio sequence: $$ r_{i} = \frac{Area(P_{i} + \epsilon)}{Area(I)}$$ Question 1: Fix $\epsilon$, what is the minimum number $N$ so that $P_{N}$ fills in all of $I$? Filling in means the ratio $r_{i} = 1$. How does the area ratio grow as $\epsilon$ increases?
Question 2: What if we replace space filling curves/trees by billiard? What if we replace billiard by a random trajectory? By random trajectory, I mean a point is randomly initialised on $\partial I$, and it follows a random uniform direction towards the interior of $I$. When the particle hits the boundary, we repeat the random direction process. In this case the area filling may be formulated probabilistically.
