On the existence of large subspaces of $C(K)$ that perform stable phase retrieval

TL;DR

This paper investigates conditions under which large subspaces of continuous functions on a compact space perform stable phase retrieval, linking the existence of such subspaces to the topological structure of the space.

Contribution

It establishes a precise criterion for the existence of infinite-dimensional stable phase retrieval subspaces in $C(K)$ based on the Cantor-Bendixson derivatives of $K$ and provides methods to construct large such subspaces.

Findings

Existence of infinite-dimensional SPR subspaces iff $K''$ is nonempty.

Construction of large SPR subspaces depending on the number of Cantor-Bendixson derivatives.

Characterization of the size of SPR subspaces in relation to the topology of $K$.

Abstract

The purpose of this article is to address an open problem posed by Freeman-Oikhberg-Pineau-T.~(\textit{Math.~Ann.}~2024) regarding the existence of large subspaces of $C(K)$ that perform stable phase retrieval (SPR). We begin by proving that for both the real and complex fields, the space $C(K)$ admits an infinite-dimensional SPR subspace if and only if the second Cantor-Bendixson derivative $K{''}$ is nonempty. We then show how to construct ``large" SPR subspaces of $C(K)$, where the size of the subspace depends quantitatively on the number of non-trivial Cantor-Bendixson derivatives that the compact Hausdorff space $K$ possesses.