Measure decay for porous sets

Statement

Let $X\subseteq\mathbb R^d$ be $\nu$ -porous on balls from scales $\alpha_0$ to $\alpha_1$ . There are constants $C,\gamma>0$ , depending only on $\nu$ and $d$ , such that every ball $B$ of radius $R$ with $\alpha_0<R<\alpha_1$ satisfies

|X\cap B|\le C R^d\left(\frac{\alpha_0}{R}\right)^\gamma.

Mechanism

Choose an $L$ -adic grid with $L\asymp_d\nu^{-1}$ . Ball porosity gives box porosity, so at every admissible depth at least one child of each occupied cube is empty. Iterating the factor $1-L^{-d}$ through $N\asymp\log(R/\alpha_0)$ levels gives the power saving.

Quantitative exponent

One may choose $\gamma$ comparable from below to $\nu^d/|\log\nu|$ , with a dimension-dependent constant. Sharpness of this particular expression is not asserted.

References

Alex Cohen, “Fractal uncertainty in higher dimensions,” 2024. arXiv record. Relevant: Lemmas A.6–A.7.