# What are the subsets of the unit circle that can be the points in which a power series is convergent?

Let $A\subset\Bbb C$ be a subset of the unit circle. Consider the following condition on $A$.

Cond. There exists a sequence $\{a_i\}_{i=1}^\infty$ of complex numbers such that $$\sum_{n=1}^\infty a_nz^n$$ is a power series with radius of convergence $1,$ and $A$ is exactly the subset of the unit circle in which the series converges.

Are there any interesting conditions on a subset $A$ of the unit circle which imply, are implied by or are equivalent to Cond.? I think all finite subsets of the circle have this property. What about the countable subsets? Does it have anything to do with measurability?

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