Approximate subgroups with bounded VC-dimension

We combine the fundamental results of Breuillard, Green, and Tao (Publ Math Inst Hautes Études Sci 116:115–221, 2012) on structure approximate groups, together with “tame” arithmetic regularity methods based work authors Terry (J Eur Soc (JEMS) 24(2):583–621, 2022), to give a theorem for finite subsets A arbitrary groups G where has “small tripling” bounded VC-dimension: Roughly speaking, up small error, will be union number translates coset nilprogression rank step (see Theorem 2.1). also prove stronger result in setting exponent 2.2). Our extend recent Martin-Pizarro, Palacín, Wolf (Selecta (N.S.) 27(4):Paper No. 53, 19,2021) stable sets tripling.