As we all know, a power series converges within its radius of convergence. Usually the problem is to understand what happens on the boundary.
We can converge on all of the boundary, as for example $\sum_{k\geq 0} \frac{z^k}{1+k^2}$ does. Or we can diverge on all of the boundary, as for example $\sum_{k\geq 0} k^2 z^k$ does.
My question is now:
If we pick a set $A\subseteq S^1 \subseteq \mathbb{C}$. Can we always find a power series $\sum_{k\geq 0} a_k z^k$ with radius of convergence equal $1$ such that $\sum_{k\geq 0} a_k x^k$ converges for $x\in A$ and diverges for $x\in S\setminus A$?
