Title: Phase retrieval on circles and lines

Abstract: Let $f$ and $g$ be analytic functions on the open unit disc $\mathbb D$ such that $|f|=|g|$ on a set $A$. We first prove 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 $\pi$. 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 the previous results and analyse the case $A=r\mathbb T$. Finally we examine the situation when there is equality on two distinct circles in the disc, proving a result or counterexample for each possible configuration.