Abstract:
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 $\Gamma$.
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