We give a structure theorem for finite subsets A of a group G where A is both NIP and an approximate subgroup (more generally with small tripling), with fixed bounds on both the NIP and tripling. This combines the results of Breuillard, Green and Tao on the structure of finite approximate subgroups of arbitrary groups up to commensurability, with the arithmetic regularity results of Conant, Pillay, and Terry on NIP subsets of finite groups.

This is joint work with Gabriel Conant. The work was supported by NSF grants DMS-1665035, DMS-1760212, and DMS-1855503