Example of a unit circle subset that is $F_{\sigma\delta}$ but not $F_\sigma$

Question

I am looking for an example of a unit circle subset that is $F_{\sigma\delta}$ but not $F_\sigma$. The task is connected with studying convergence of power series on the boundary of the convergence set, so this is why being a subset of circle is needed.

Is $\{z\in\mathbb{C}:|z|=1\}- \{z\in\mathbb{C}:z=e^{i\pi\phi}, \phi\in\mathbb{Q}\}$ a good example?

Answer 1

Yes, that seems right. It's not $F_{\sigma}$ by the argument given in answer to this question and it's $F_{\sigma\delta}$ because it equals $$\bigcap_{q\in \mathbf Q}S^1\setminus\{e^{i\pi q}\}$$