Box porosity - Knowlpedia

Definition

Fix an integer $L\ge3$ . Partition $[-1,1]^d$ at depth $n$ into congruent cubes of side length proportional to $L^{-n}$ . A set $X\subseteq[-1,1]^d$ is box porous at scale $L$ with depth $n$ if every depth- $n$ cube $Q$ meeting $X$ contains a depth-(n+1) child cube $Q'\subset Q$ with $Q'\cap X=\varnothing$ .

Iterated form

If the condition holds at depths $0,\ldots,N-1$ , then at least one of the $L^d$ children is discarded at every occupied node. Consequently

|X|\le 2^d(1-L^{-d})^N,

where $|X|$ denotes Lebesgue measure.

Comparison with ball porosity

A set that is $\nu$ -porous on balls down to scale $h$ is box porous at all depths with $L^{-n}\gtrsim h$ , for a choice $L\asymp_d\nu^{-1}$ . Box porosity is useful because the preceding counting estimate turns geometric holes into a power-law volume bound.

References

Rui Han and Wilhelm Schlag, “A higher-dimensional Bourgain–Dyatlov fractal uncertainty principle,” Analysis & PDE 13 (2020), 813–863. DOI record.
Alex Cohen, “Fractal uncertainty in higher dimensions,” 2024. arXiv record. Relevant: Appendix A.1.