Long lines in subsets of large measure in high dimension

TL;DR

This paper proves that large measure subsets in high-dimensional spaces contain lines with significantly large intersections, and explores how this behavior varies across different measures and geometric settings.

Contribution

It establishes tight bounds on the existence of long lines in high measure sets and analyzes phase transitions for various measures, including product measures and $ ext{l}_p$ balls.

Findings

Existence of lines with intersection measure at least $ ext{Omega}(n^{1/4})$ in high measure sets

Unified behavior for a broad class of product measures

Phase transitions in intersection properties for $ ext{l}_p$ balls

Abstract

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 $\Omega ( n^{1/4} )$. The exponent $1/4$ is tight. More generally, for a probability measure $\mu $ on $\mathbb R ^n$ and $0<a<1$ define \begin{equation*} L(\mu ,a):= \inf_{A ; \mu(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 $L(\mu ,a)$ when $\mu $ is a product measure and when $\mu $ 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.