AI 中文总结
本文证明有向均匀超图的低差异等价于谱间隙，通过建立基于变分特征值的Frieze-Kannan型张量谱正则引理，给出Lenz-Mubayi定理的简短证明，并讨论相关变体。
AI 中文摘要
我们证明对于有向$k$-均匀超图，低差异等价于谱间隙。对于无向超图，这恢复了Lenz和Mubayi的一个定理，且证明显著更短。我们论证的核心是基于Friedman--Wigderson的超图特征值变分概念，为超图建立了一个Frieze--Kannan型谱正则引理，该引理将任何张量分解为有界数量的秩一张量加上一个具有小最大特征值的拟随机张量。此引理可能具有独立意义，我们对其进行了证明，适用于复值且不一定对称的张量。我们还简要讨论了Szemerédi型变体以及有限阿贝尔群上Cayley型超图的正则化。
英文摘要
We show that low discrepancy is equivalent to spectral gap for directed $k$-uniform hypergraphs. For undirected hypergraphs this recovers a theorem of Lenz and Mubayi, with a considerably shorter proof. At the heart of our argument is a Frieze--Kannan type spectral regularity lemma for hypergraphs based on the variational notion of hypergraph eigenvalues by Friedman--Wigderson, which decomposes any tensor into a bounded number of rank-one tensors plus a quasi-random tensor with small top eigenvalue. This lemma may be of independent interest, and we prove it for complex-valued, not necessarily symmetric tensors. We also briefly discuss a Szemerédi-type variant and the regularization of Cayley-type hypergraphs over finite abelian groups.
Comments15 pages
