论文标题
高维扩展器中的双平衡套件
Double Balanced Sets in High Dimensional Expanders
论文作者
论文摘要
最近的作品表明，伪随机集的扩展非常重要。但是，伪随机集的所有当前作品仅限于可以应用傅立叶分析方法的产品（或近似产品）空间。在这项工作中，我们询问了一个自然的问题，即在无法应用傅立叶分析方法（例如单方面局部频谱扩展器）的域中是否相关。 我们在回答这个问题的道路上迈出了第一步。我们提出了一个新的定义，该定义为``双重平衡套件''。我们通过证明单方面局部光谱扩展器中的小双平衡集具有非常强大的扩展属性，例如独特的邻居状扩展，从而证明了我们的新定义的强度。我们进一步表明，宇宙扩张器中的共同体是双平衡的，并使用新衍生的双平衡集合的强大扩展特性，以获得对最小距离的最低距离的最低限制的当前状态的指数改进。
Recent works have shown that expansion of pseudorandom sets is of great importance. However, all current works on pseudorandom sets are limited only to product (or approximate product) spaces, where Fourier Analysis methods could be applied. In this work we ask the natural question whether pseudorandom sets are relevant in domains where Fourier Analysis methods cannot be applied, e.g., one-sided local spectral expanders. We take the first step in the path of answering this question. We put forward a new definition for pseudorandom sets, which we call ``double balanced sets''. We demonstrate the strength of our new definition by showing that small double balanced sets in one-sided local spectral expanders have very strong expansion properties, such as unique-neighbor-like expansion. We further show that cohomologies in cosystolic expanders are double balanced, and use the newly derived strong expansion properties of double balanced sets in order to obtain an exponential improvement over the current state of the art lower bound on their minimal distance.
