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Let $\mu$ be the logarithmic equilibrium measure on a compact set $\gamma \subset {\mathbb{R}}^d$. We prove that $\mu$ is absolutely continuous with respect to the length measure on the part of $\gamma$ which can be locally expressed as the graph of a …

The Favard length of a planar Borel set is the average length of its orthogonal projections. We prove that if an Ahlfors 1-regular set has large Favard length, then it contains a big piece of a Lipschitz graph. This gives a quantitative version of …

Let $s\in [0,1]$. We show that a Borel set $N\subset {\mathbb{R}}^2$ whose every point is linearly accessible by an $s$-dimensional family of lines has Hausdorff dimension at most $2-s$.