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?
 A: Try this POST or, indeed, the whole thread.  A quote:

The convergence set has to be F_sigma_delta, since
  the (pointwise) convergence set for any sequence
  of continuous functions is F_sigma_delta.
Herzog and Piranian (together) proved in 1949 that
  any F_sigma subset of |z| = 1 can be the convergence set
  of some power series with radius of convergence 1. Lukasenko
  proved in 1978 that some G_delta subsets of |z| = 1 cannot
  be the convergence set of any power series with radius of
  convergence 1. For a fairly elementary survey of the problem
  of characterizing the convergence set for a power series
  in C (complex numbers), see Thomas W. Korner, "The
  behavior of power series on their circle of convergence"
  [pp. 56-94 in "Banach Spaces, Harmonic Analysis, and
  Probability Theory", Springer Lecture Notes in Mathematics
  995, Springer-Verlag, 1983]. This is a beautifully written
  paper that contains detailed proofs of virtually everything
  and is pitched at the level of a beginning graduate student
  in math.
Dave L. Renfro

