Remarks on the descriptive metric characterization of singular sets of analytic functions

This work presents two remarks on the structure of singular boundary sets of functions analytic in the unit disk $D$: $|z|<1$. The first remark concerns the conversion of the Plessner theorem. We prove that three pairwise disjoint subsets $E_1,E_2$, and $E_3$ of the unit circle $\Gamma$: $|z|=1$, $\bigcup_{i=1}^3E_i=\Gamma$ are the sets $I(f)$ of all Plessner points, $F(f)$ of all Fatou points, and $E(f)$ of all exceptional boundary points, respectively, for a function $f$ holomorphic in $D$ if and only if $E_1$ is a $G_\delta$-set and $E_3$ is a $G_{\delta\sigma}$-set of linear measure zero. In the second part of the paper it is shown that for any $G_{\delta\sigma}$-subset $E$ of the unit circle $\Gamma$ with a zero logarithmic capacity there exists a one-sheeted function on $D$ whose angular limits do not exist at the points of $E$ and do exist at all the other points of $\Gamma$.