Porosity on lines - Knowlpedia

Definition

Let $0<\nu\le 1$ and $0<\alpha_0<\alpha_1$ . A set $Y\subseteq\mathbb R^d$ is $\nu$ -porous on lines from scales $\alpha_0$ to $\alpha_1$ if, for every line segment $\tau$ of length $R$ with $\alpha_0<R<\alpha_1$ , some $x\in\tau$ satisfies

B_{\nu R}(x)\cap Y=\varnothing.

The missing ball may extend away from the line, but its center must lie on the tested segment.

Why this is stronger than ball porosity

Every line-porous set is porous on balls, after an inessential adjustment of constants. The converse fails for $d\ge2$ : a straight line has large holes inside ambient balls but has none along segments lying in that line. Line porosity therefore excludes orthogonal linear concentrations that obstruct higher-dimensional uncertainty estimates.

Stability properties

Dilating $Y$ by $s>0$ multiplies both endpoint scales by $s$ and preserves $\nu$ . Intersecting with any fixed line produces a one-dimensional porous set. A sufficiently small Minkowski thickening remains line porous with a smaller porosity constant and a larger lower scale.

References

Alex Cohen, “Fractal uncertainty in higher dimensions,” 2024. arXiv record. Relevant: §§1.2 and A.2.