Examples of Taylor series with interesting convergence along the boundary of convergence?

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.

Solution 1:

[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.