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In most standard examples of power series, the question of convergence along the boundary of convergence has one of several "simple" answers. (I am considering power series of a complex variable.)

  • The series always converges along its boundary $\left(\displaystyle\sum_{n=1}^\infty\frac{x^n}{n^2}\right)$
  • The series never converges along its boundary $\left(\displaystyle\sum_{n=0}^\infty\ x^n\right)$
  • The series diverges at precisely one point along its boundary $\left(\displaystyle\sum_{n=1}^\infty\ \frac{x^n}{n}\right)$
  • The series diverges at a finite number of points along its boundary (add several examples of the preceding type together).

Can anything else happen?

  1. For starters, are there examples where the series converges at precisely a finite number of points along the boundary?
  2. Can there be a dense mixture of convergence and divergence along the boundary? For example, maybe a series with radius $1$ that converges for $x=\mathrm{e}^{2\pi it}$ with $t$ rational, but diverges when $t$ is irrational.
  3. Can there be convergence in large connected regions on the boundary with simultaneous divergence in other large connected regions? For example, a series with radius $1$ that converges along the "right side" of the boundary $\left(x=\mathrm{e}^{2\pi it}\mbox{ with }t\in\left(-\frac{1}{4},\frac{1}{4}\right)\right)$ and diverges elsewhere on the boundary.

I'm curious for any examples of these types or any type beyond the bulleted types.

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1 Answer 1

up vote 14 down vote accepted

[Edit (Jan.12/12): Updated in light of the reference mentioned by Dave Renfro in the comments below.]

This is a very interesting topic. I wrote a somewhat longish answer on it for a similar question that came up on MathOverflow a bit ago. In there (and the comments) you will find references, and links to most of the relevant papers.

The short version is that any possible "set of convergence" is an $F_{\sigma\delta}$ subset of the boundary, but not all such sets are possible. We know that every $F_\sigma$ set is possible, and a bit more, but Körner has shown that some $G_\delta$ sets are not sets of convergence.

A particular open problem that has received some attention is whether every $F_{\sigma\delta}$ set of full measure is of convergence.

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Unless I missed it, the following reference is not mentioned, and it's something the original poster may find more useful than the original papers: 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 survey of characterizing the convergence set for a power series in $\mathbb C$, containing detailed proofs of virtually everything and pitched at the level of a beginning graduate student in math. –  Dave L. Renfro Nov 17 '11 at 15:55
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I should mention that one of the drawbacks to Korner's survey is that he gives almost no references or historical comments (maybe none at all -- it's been several years since I last looked at it and I don't remember now). My guess is that his survey was lightly edited from class or seminar notes, and maybe the preface says something to this effect (again, I don't remember). As for giving super-detailed and complete proofs, his survey is similar to C. A. Rogers' book Hausdorff Measures, except the Rogers book gives a lot of references. –  Dave L. Renfro Nov 17 '11 at 15:56
    
Dear Dave, Many thanks! I wasn't aware of this paper (unfortunate that, as you say, it has no references). –  Andres Caicedo Nov 17 '11 at 16:21
    
Thanks Andres and Dave. I'm going to look these up as soon as I get a chance. I'd like to see an explicitly defined example with interesting behavior. Existence proofs or constructions that feel too indirect will leave me wanting more, but it sounds like there will be "direct" examples in these sources. –  alex.jordan Nov 17 '11 at 22:22

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