Bounds For Multilinear Sublevel Sets Via Szemeredi's Theorem

TL;DR

This paper establishes near-complete bounds for multilinear sublevel sets using an extension of Szemeredi's theorem, advancing understanding of oscillatory integral operators under certain rationality conditions.

Contribution

It provides the first broad sublevel set bounds for multilinear operators, extending Szemeredi's theorem to this context under rationality assumptions.

Findings

Sublevel set bounds are established in a weak form.

The bounds are nearly general, subject to a rationality hypothesis.

The proof extends Szemeredi's theorem to multilinear settings.

Abstract

In 2005, Li, Tao, Thiele and the author raised a general question concerning upper bounds for a class of multilinear oscillatory integral operators, and established such bounds in a few cases. Most cases remain open. The present paper is concerned with sublevel set bounds, which would be a consequence of the oscillatory integral bounds, if valid. These sublevel set bounds are established here in a weak form but in nearly full generality, subject only to a rationality hypothesis. The proof relies on an extension of Szemeredi's theorem due to Furstenberg and Katznelson.