# 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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## 1 Answer

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

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