Thin set

Summary

A set $E \subseteq \R^n$ is thin at a point $x$ if $E$ is very sparse near $x$. Precisely, some superharmonic function is strictly larger along $E$ near $x$ than at $x$ itself. Thinness measures how visible a set is to Brownian motion and to harmonic functions near a point. A boundary point of a domain is irregular for the Dirichlet problem exactly when the complement of the domain is thin there. Wiener's criterion decides thinness by a series of capacities.