Stable phase retrieval in function spaces

   Let $(\Omega,\Sigma,\mu)$ be a measure space and $1\leq p\leq \infty$. A subspace $E\subseteq L_p(\mu)$ is said to do \emph{stable phase retrieval (SPR)} if there exists a constant $C\geq 1$ such that for any $f,g\in E$ we have 
    \begin{equation}
       \inf_{|\lambda|=1} \|f-\lambda g\|\leq C\||f|-|g|\|.
    \end{equation}
    In this  case, if $|f|$ is known, then $f$ is uniquely determined up to an unavoidable global phase factor $\lambda$; moreover, the phase recovery map is $C$-Lipschitz. Phase retrieval appears in several applied circumstances, ranging from crystallography to quantum mechanics.

In this talk, I will present some elementary examples of subspaces of $L_p(\mu)$ which do stable phase retrieval and discuss the structure of this class of subspaces. In particular, I will explain how SPR connects to $\Lambda(p)$-set theory, which is a classical topic in the intersection of number theory and harmonic analysis. 

The material in this talk is based on joint work with M.~Christ and B.~Pineau as well as with D.~Freeman, T.~Oikhberg and B.~Pineau.