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–δ 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 3/4.

Our work is motivated by [DKK⁺18], wherein the authors show that a linearity agreement hypothesis implies an NP-hardness gap of 1/2–ε vs. ε for Unique Games and other inapproximability results. Barak, Kothari and Steurer show that the hypothesis in this work implies the linearity agreement hypothesis [DKK⁺18].

Following initial publication of this work, our hypothesis was proved in [KMS18].

我们研究了 Grassmann 图中非扩展集的结构。我们提出了一个假设，说明每个扩展小于 1- δ 的小集合必须与指定的一组集合中的一个相关，这些集合与较小的 Grassmann 图同构。我们开发了一个傅里叶分析框架来分析 Grassmann 图上的函数，并证明我们的假设适用于所有展开低于 3/4 的集合。

我们的工作受到 [DKK ⁺ 18] 的启发，其中作者表明，线性一致性假设意味着 1/2– ε与ε的 NP 硬度差距对于 Unique Games 和其他不可近似性结果。Barak、Kothari 和 Steurer 表明，这项工作中的假设意味着线性一致性假设 [DKK ⁺ 18]。

在这项工作最初发表后，我们的假设在 [KMS18] 中得到了证明。