Abstract.

There are many subsets $N$ of $\mathbb{R}^n$ for which one can find a real-valued Lipschitz function $f$ defined on the whole of $\mathbb{R}^n$ but non-differentiable at every point of $N$. Of course, by the Rademacher theorem any such set $N$ is Lebesgue null, however, due to a celebrated result of Preiss from 1990 not every Lebesgue null subset of $\mathbb{R}^n$ gives rise to such a Lipschitz function $f$. In this talk, I explain that a sufficient condition on a set $N$ for such $f$ to exist is to be locally unrectifiable with respect to curves in a cone of directions. In particular, every $1$-purely unrectifiable set $U$ possesses a Lipschitz function non-differentiable on $U$ in the strongest possible sense. I also give an example of a universal differentiability set unrectifiable with respect to a fixed cone of directions, showing that one cannot relax the conditions. This is joint work with D. Preiss.