论文标题
高度大小的子集中的长线
Long lines in subsets of large measure in high dimension
论文作者
论文摘要
我们证明，对于任何集合$ a \ subseteq [0,1]^n $，带有$ \ text {vol}（a）\ ge 1/2 $，存在一条线$ \ ell $，使得一维的lebesgue量$ \ ell \ cap a $至少为$ω（N^{1/4}）$。指数$ 1/4 $很紧。更一般而言，对于概率度量，$ \ Mathbb r ^n $和$ 0 <a <1 $ define \ begin {equination*} l（μ，a）：= \ inf_ {a; μ（a）= a} \ sup _ {\ ell \ text {line}} \ big | \ ell \ cap a \ big | \ end {equation*}其中$ | \ cdot | $代表一维的Lebesgue度量。当$μ$是产品量度时，当$μ$是$ \ ell _p $ ball上的均匀度量时，我们研究$ L（μ，a）$的渐近行为。我们在大量产品测量中观察到了相当统一的行为。另一方面，对于$ 1 \ leq p \ leq \ infty $的$ \ ell_p $ balls，我们发现有不同类型的相变。
We show that for any set $A\subseteq [0,1]^n$ with $\text{Vol}(A)\ge 1/2$ there exists a line $\ell $ such that the one-dimensional Lebesgue measure of $\ell \cap A$ is at least $Ω( n^{1/4} )$. The exponent $1/4$ is tight. More generally, for a probability measure $μ$ on $\mathbb R ^n$ and $0<a<1$ define \begin{equation*} L(μ,a):= \inf_{A ; μ(A) = a} \sup _{\ell \text{ line}} \big| \ell \cap A\big| \end{equation*} where $|\cdot | $ stands for the one-dimensional Lebesgue measure. We study the asymptotic behavior of $L(μ,a)$ when $μ$ is a product measure and when $μ$ is the uniform measure on the $\ell _p$ ball. We observe a rather unified behavior in a large class of product measures. On the other hand, for $\ell_p$ balls with $1 \leq p \leq \infty$ we find that there are phase transitions of different types.
