Herglotz's theorem is a foundational result in functional analysis, establishing that the matrix coefficients of the powers of a unitary operator can be represented as integrals of linear phases against a complex-valued spectral measure on the unit circle.

When the unitary operator in question is a Koopman operator, these matrix coefficients can be naturally extended to multilinear forms, known as multicorrelation sequences, which play a fundamental role in additive combinatorics (most notably serving as the core machinery behind Furstenberg's landmark proof of Szemerédi's theorem).

In this talk we will explore recent progress, open conjectures and obstacles surrounding a potential generalization of Herglotz's theorem for these multicorrelation sequences.