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.
Cutting lemma and Zarankiewicz’s problem in distal structures.Selecta Mathematica, New Series, 26(2):8, 2020
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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.