Line-section decay for line-porous sets

Statement

Let $Y\subseteq\mathbb R^d$ be $\nu$ -porous on lines from scales $\alpha_0$ to $\alpha_1$ . There are $C,\gamma>0$ , depending only on $\nu$ , such that every line segment $\tau$ of length $R$ , with $\alpha_0<R<\alpha_1$ , satisfies

\mathcal L^1(\tau\cap Y) \le C R\left(\frac{\alpha_0}{R}\right)^\gamma.

Here $\mathcal L^1$ is one-dimensional Lebesgue measure on the supporting line.

Proof idea

The intersection of $Y$ with its supporting line is a one-dimensional porous set. Applying measure decay for porous sets in dimension one yields the estimate.

Role in fractal uncertainty

The estimate controls integrals of weights along all lines. That is precisely the geometry measured by the radial-line growth functional in the higher-dimensional Beurling–Malliavin theorem.

References

Alex Cohen, “Fractal uncertainty in higher dimensions,” 2024. arXiv record. Relevant: Corollary A.8.