论文标题
连接的非铸群的最小值和几乎最小的测量膨胀
Minimal and nearly minimal measure expansions in connected unimodular groups
论文作者
论文摘要
令$ g $为一个配备A（左侧为右）HAAR度量$μ_g$的连接的单型组，并假设$ a，b \ subseteq g $是非空的且紧凑的。 Kemperman的不平等给我们$μ_g（AB）\ geq \ min \ {μ_g（a）+μ_g（b），μ_g（g）\}。 我们的第一个结果决定了平等的条件，为肯珀曼（Kemperman）在1964年提出的一个问题提供了完整的答案。我们的第二个结果表征紧凑和连接的$ g $，$ a $和$ b $几乎实现平等，具有尖锐指数的定量界限。可以将其视为$（3K-4）$ - 定理此设置的定理，并确认Griesmer和Tao的猜想相关案例。作为应用程序，我们获得了连接的紧凑型谎言组的度量扩展差距结果。 我们的证明中开发的工具包括分析最小和几乎最小扩展的一组形状的分析，从此桥梁到某个伪计的属性，以及适当的连续组同构的构建，以构建$ \ MATHBB {r} $或$ \ Mathbb {t} = \ Mathbb {t）
Let $G$ be a connected unimodular group equipped with a (left and hence right) Haar measure $μ_G$, and suppose $A, B \subseteq G$ are nonempty and compact. An inequality by Kemperman gives us $μ_G(AB)\geq\min\{μ_G(A)+μ_G(B),μ_G(G)\}.$ Our first result determines the conditions for the equality to hold, providing a complete answer to a question asked by Kemperman in 1964. Our second result characterizes compact and connected $G$, $A$, and $B$ that nearly realize equality, with quantitative bounds having the sharp exponent. This can be seen up-to-constant as a $(3k-4)$-theorem for this setting and confirms the connected case of conjectures by Griesmer and by Tao. As an application, we get a measure expansion gap result for connected compact simple Lie groups. The tools developed in our proof include an analysis of the shape of minimally and nearly minimally expanding pairs of sets, a bridge from this to the properties of a certain pseudometric, and a construction of appropriate continuous group homomorphisms to either $\mathbb{R}$ or $\mathbb{T} = \mathbb{R}/\mathbb{Z}$ from the pseudometric.
