# An application of Grothendieck theorem to the theory of multicorrelation sequences, multiple recurrence and partition regularity of quadratic equations

## TL;DR

This paper applies Grothendieck's theorem to analyze multicorrelation sequences and derive new results in multiple recurrence and partition regularity of quadratic equations, extending previous work in ergodic theory and combinatorics.

## Contribution

It introduces a structure theorem for multicorrelation sequences using Grothendieck's theorem, leading to novel recurrence and partition regularity results for quadratic systems.

## Key findings

- Established a structure theorem for multicorrelation sequences of length two.
- Proved a multiple recurrence result for products of linear terms.
- Demonstrated partition regularity of certain quadratic equations.

## Abstract

We use Grothendieck theorem to prove a structure theorem for multicorrelation sequences of length two, associated with two (not necessarily commuting) measure preserving actions on a probability space. We use this to deduce a multiple recurrence result concerning products of linear terms, and a partition regularity result of certain systems of quadratic equations, building on the work of Frantzikinakis and Host.