Description
We show that there is $0<\alpha_0<1$ depending on the parameters of the percolation such that the fractal (Mandelbrot) percolation is almost surely purely $\alpha$-unrectifiable for all $\alpha>\alpha_0$.
For a $0 < \alpha \le 1$ a set $A \in \mathbb{R}^d$ is purely $\alpha$-unrectifiable, if
$H^{1/\alpha} (A \cap \gamma([0, 1])) = 0$ holds for every $\alpha$-Hölder curve $\gamma \colon [0, 1] \to \mathbb{R}^d$, where $H^s$ denotes the $s$-dimensional Hausdorff measure.
We consider fractal percolation based on a subdivision of $[0,1]^d$ into
$N^d$ many subcubes.
We show that for every $d=2,3,\dots$, $0 \le p < 1$ and $N$ there exists $\alpha_0 < 1$ such that the fractal percolation set $E$ is almost surely purely $\alpha$-unrectifiable for all $\alpha>\alpha_0$.
Since the case $\alpha = 1$ corresponds to standard $1$-unrectifiability, this result implies that $E$ is almost surely purely $1$-unrectifiable.
In our paper a simpler proof than that of our main theorem is given for $1$-unrectifiability.
The seventh section of our paper contains new tools that we develop for the general case. We believe that these new tools turn out to be useful in many other problems
related to the fractal percolation and other random geometric constructions.
The talk is based on a joint paper with Esa Järvenpää, Maarit Järvenpää, Tamás Keleti and Tuomas Pöyhtäri.
Zoom link: https://zoom.us/j/93746696898?pwd=b1J2MnEwMVdDVElPUFRkYWdtVXdWdz09
Meeting ID: 937 4669 6898
Passcode: 280561