I asked myself the following question while preparing a course on power series for 2nd year students. Let $F$ be the set of power series with convergence radius equal to $1$. What subsets $S$ of the unit circle $C$ can be realised as $$ S:=\{x \in C: f\text{ diverges in }x\} $$ for $f \in F$? Any finite subset (and possibly any countable subset) of $C$ can be realised that way. Who knows more on this?