Quantitative structure of stable sets in arbitrary finite groups

We show that a k k -stable set in finite group can be approximated, up to given error alttext="epsilon greater-than 0"> ϵ > 0 encoding="application/x-tex">\epsilon >0 , by left cosets of subgroup index Superscript hyphen upper O Super Subscript k left-parenthesis 1 right-parenthesis"> - O ( 1 stretchy="false">) ^{\text {-} O_k(1)} . This improves the bound similar result Terry and Wolf on stable arithmetic regularity abelian groups, leads quantitative account work author, Pillay, sets arbitrary groups. also prove an analogous for small tripling which provides version recent Martin-Pizarro, Palacín, Wolf. Our proofs use results VC-dimension, finitization model-theoretic techniques from theory.