Phase retrieval on circles and lines

Let $f$ and $g$ be analytic functions on the open unit disc $\mathbb D$ such that $|f|=|g|$ on a set $A$. We give an alternative proof of the result of Perez that there exists $c$ in the unit circle $\mathbb T$ such that $f=cg$ when $A$ is the union of two lines in $\mathbb D$ intersecting at an angle that is an irrational multiple of $\pi$, and from this deduce a sequential generalization of the result. Similarly, the same conclusion is valid when $f$ and $g$ are in the Nevanlinna class and $A$ is the union of the unit circle and an interior circle, tangential or not. We also provide sequential versions of this result and analyse the case $A=r\mathbb T$. Finally, we examine the most general situation when there is equality on two distinct circles in the disc, providing a result or counterexample for each possible configuration.