Phase retrieval on circles and lines

Let f and g be analytic functions on the open unit disk 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 T such that $f=cg$ when A is the union of two lines in D intersecting at an angle that is an irrational multiple of π , 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 analyze the case A=r T . Finally, we examine the most general situation when there is equality on two distinct circles in the disk, proving a result or counterexample for each possible configuration.