On non-optimally expanding sets in grassmann graphs

We study the structure of non-expanding sets in the Grassmann graph. We put forth a hypothesis stating that every small set whose expansion is smaller than 1 -d must be correlated with one of a specified list of sets which are isomorphic to smaller Grassmann graphs. We develop a framework of Fourier analysis for analyzing functions over the Grassmann graph, and prove that our hypothesis holds for all sets whose expansion is below 7/8. In the companion submitted paper [Dinur, Khot, Kindler, Minzer and Safra, STOC 2018], the authors show that a linearity agreement hypothesis implies an NP-hardness gap of 1/2 - epsilon vs epsilon for unique games and other inapproximability results. In [Barak, Kothari and Steurer, ECCC TR18-077], the authors show that the hypothesis in this work implies the linearity agreement hypothesis of [Dinur, Khot, Kindler, Minzer and Safra, STOC 2018]. Combined with our main theorem here this proves a version of the linearity agreement hypothesis with certain specific parameters. Short of proving the entire hypothesis, this nevertheless suffices for getting new unconditional NP hardness gaps for label cover with 2-to-1 and unique constraints. Our Expansion Hypothesis has been subsequently proved in its full form [Khot, Minzer and Safra, ECCC TR18-006] thereby proving the agreement hypothesis of [Dinur, Khot, Kindler, Minzer and Safra, STOC 2018] and completing the proof of the 2-to-1 Games Conjecture (albeit with imperfect completeness).