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?