Porosity on balls - Knowlpedia

Definition

Let $0<\nu\le 1$ and $0<\alpha_0<\alpha_1$ . A set $X\subseteq\mathbb R^d$ is $\nu$ -porous on balls from scales $\alpha_0$ to $\alpha_1$ if every Euclidean ball $B$ whose diameter $R$ satisfies $\alpha_0<R<\alpha_1$ contains a point $x\in B$ such that

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

Thus every admissible observation ball contains a hole whose radius is a fixed fraction of the observation scale.

Scale dependence

Porosity here is explicitly truncated: nothing is required below $\alpha_0$ or above $\alpha_1$ . This is the form used in quantitative fractal uncertainty principles, where the smallest scale is tied to a semiclassical parameter.

Relation to other notions

Porosity on lines tests every line segment and is stronger in dimensions at least two. In one dimension the two definitions agree up to the harmless convention of using radius or diameter as the scale. Porosity also forces quantitative decay of Lebesgue measure through the porous-set measure-decay estimate.

References

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