On the sharpness of certain approach regions

In this survey, we describe joint work in collaboration with A. Stokolos, O. Svensson and T. Weiss. We consider the following question: How sharp is the Stolz approach region condition for the almost everywhere convergence of bounded harmonic functions? The issue was first settled in the rotation invariant case in the unit disc by Littlewood in 1927 and later examined, under less stringent conditions, by Aikawa in 1991. We show that our results are, in a precise sense, sharp.