高维平坦环面上Lipschitz函数的振动刚性
On the oscillation rigidity of a Lipschitz function on a high-dimensional flat torus
摘要：
给定环面$\mathbb{T}^n $上的任意$1$-Lipschitz函数$f$，我们找到一个$k$维的子环面$M \subseteq \mathbb{T}^n$，它平行于轴，使得$f$对子环面$M$的约束几乎是一个常数函数.随机均匀地选择$k$维子环面$M$。我们表明，当$k \leq c \log n / (\log \log n + \log 1/\varepsilon)$，最大值和最小值的$f$在这个随机子环面$M$不同的最多$\varepsilon$，具有高概率。
英文摘要：
Given an arbitrary $1$-Lipschitz function $f$ on the torus $\mathbb{T}^n $, we find a $k$-dimensional subtorus $M \subseteq \mathbb{T}^n$, parallel to the axes, such that the restriction of $f$ to the subtorus $M$ is nearly a constant function. The $k$-dimensional subtorus $M$ is chosen randomly and uniformly. We show that when $k \leq c \log n / (\log \log n + \log 1/\varepsilon)$, the maximum and the minimum of $f$ on this random subtorus $M$ differ by at most $\varepsilon$, with high probability.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
