Tangential Boundary Behavior of Bounded Harmonic Functions in the Unit Disc

Bounded harmonic functions in the unit disc $D$ converge nontangentially almost everywhere (Fatou, 1906) and fail to converge along the rotates of any given tangential curve (Littlewood, 1927). We study their boundary behaviour along tangential curves whose shape may change from point to point (a problem posed by W. Rudin). Let $\tau$ be the assignment of a curve $\tau_\theta$ in $D$ ending at $\theta$ and tangential to the boundary $bD$ of $D$, for each $\theta\in bD$. The authors announce a proof that convergence along $\tau$ fails if $\tau_\theta$ depends on $\theta$ in a measurable way, and to show that there is a family $\tau$ of tangential curves such that each bounded harmonic function in $D$ converges along $\tau_\theta$ for a set of points $\theta$ whose outer measure is equal to $2\pi$.