Stabilizers and NIP arithmetic regularity

We give a new proof of the NIP arithmetic regularity lemma for finite groups (due to the authors and Pillay), which describes the approximate structure of "NIP sets" in finite groups, i.e., subsets whose collection of left translates has bounded VC-dimension. Our new proof avoids sophisticated ingredients from the model theory of NIP formulas (e.g., Borel definability and generic compact domination). The key tool is an elaboration on an elementary lemma due to Alon, Fox, and Zhao concerning the behavior of subgroups contained in stabilizers. We adapt this lemma to arbitrary subsets of stabilizers using technical (but elementary) maneuvers based on work of Sisask. Using another trick from Alon, Fox, and Zhao, we then give an effective proof of a related result of the first author and Pillay on finite NIP sets of bounded tripling in arbitrary groups. Along the way, we show that NIP sets satisfy a strong form of the Polynomial Bogolyubov-Ruzsa Conjecture.