Strongly n-distal NIP theories admit a hypergraph regularity lemma, compact domination for definable fsg groups, and the n-distality hierarchy is strict among stable theories; infinite such fields have characteristic zero.
Structure and regularity for subsets of groups with finite VC-dimension.Journal of the European Mathematical Society, 24(2):583–621, 2021
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On n-distality, n-triviality and hypergraph regularity in NIP theories
Strongly n-distal NIP theories admit a hypergraph regularity lemma, compact domination for definable fsg groups, and the n-distality hierarchy is strict among stable theories; infinite such fields have characteristic zero.