On n-distality, n-triviality and hypergraph regularity in NIP theories

TL;DR

This paper investigates properties of Keisler measures in strongly n-distal NIP theories, establishing a hypergraph regularity lemma, compact domination, and characteristic results for fields, extending prior work in model theory.

Contribution

It generalizes results for distal theories to strongly n-distal NIP theories, introduces a hypergraph regularity lemma, and proves that infinite strongly n-distal NIP fields have characteristic zero.

Findings

Established a hypergraph version of the distal regularity lemma.

Proved compact domination for definable fsg groups.

Showed infinite strongly n-distal NIP fields have characteristic 0.

Abstract

We study Keisler measures in strongly n-distal NIP theories, generalizing some results of Simon and Chernikov-Starchenko for distal theories and addressing some questions of Walker. In particular, we establish a hypergraph version of the distal regularity lemma, compact domination for definable fsg groups, and demonstrate that the strong n-distality hierarchy is strict among stable theories using a connection to Poizat's total triviality of forking. We also show that infinite strongly n-distal NIP fields have characteristic 0 using a discrepancy result of Babai-Hayes-Kimmel from multiparty communication complexity.