论文标题
将Carnot群的雪花嵌入到有限的欧几里得空间中，具有最佳的失真
Embedding snowflakes of Carnot groups into bounded dimensional Euclidean spaces with optimal distortion
论文作者
论文摘要
我们表明，对于任何Carnot group $ g $，都有一个自然数量$ d_g $，因此对于任何$ 0 <\ varepsilon <1/2 $ <1/2 $ the metric space $（g，d_g^{1- \ varepsilon}） $ o_g（\ varepsilon^{ - 1/2}）$。这是通过基于T. Tao（2021）的方法来完成的，后者在使用Nash-Moser迭代方案的新变体中，结合了正常矢量场的新扩展定理时，建立了上述断言。除了需要克服Carnot群体更一般环境中的几个技术问题之外，我们的证据与TAO的证明相距甚远。
We show that for any Carnot group $G$ there exists a natural number $D_G$ such that for any $0<\varepsilon<1/2$ the metric space $(G,d_G^{1-\varepsilon})$ admits a bi-Lipschitz embedding into $\mathbb{R}^{D_G}$ with distortion $O_G(\varepsilon^{-1/2})$. This is done by building on the approach of T. Tao (2021), who established the above assertion when $G$ is the Heisenberg group using a new variant of the Nash--Moser iteration scheme combined with a new extension theorem for orthonormal vector fields. Beyond the need to overcome several technical issues that arise in the more general setting of Carnot groups, a key point where our proof departs from that of Tao is in the proof of the orthonormal vector field extension theorem, where we incorporate the Lovász local lemma and the concentration of measure phenomenon on the sphere in place of Tao's use of a quantitative homotopy argument.
