Purely unrectifiable sets, fractal percolation and graphs of functions

Abstract

This paper contains a survey of some of the results of the author related to unrectifiablity and is an extended version of the author&#39;s talk given at the Second Winter School Geometric Measure Theory Rectifiability vs. Pure Unrectifiability in Hanghzou, China. These results include irregular/purely unrectifiable $1$-sets on the graphs of continuous functions like the Takagi, the Weierstrass-Cellerier and the typical (in the sense of Baire) continuous function. It is also discussed that there exists $ α_{0}<1$ such that the fractal percolation is almost surely purely $α$-unrectifiable for all $α>α_0$. The background of the $1$-unrectifiability is discussed in more detail.