Higher-dimensional line-porous fractal uncertainty principle

Statement

Let $d\ge1$ , $0<\nu\le1/3$ , and $0<h<1/100$ . Suppose $X\subseteq[-1,1]^d$ is $\nu$ -porous on balls from scales $h$ to $1$ , and $Y\subseteq[-h^{-1},h^{-1}]^d$ is $\nu$ -porous on lines from scales $1$ to $h^{-1}$ . Then constants $C,\beta>0$ , depending only on $\nu$ and $d$ , satisfy

\operatorname{supp}\widehat f\subseteq Y \quad\Longrightarrow\quad \|f\mathbf1_X\|_2\le Ch^\beta\|f\|_2

for every $f\in L^2(\mathbb R^d)$ .

Proof architecture

Line porosity produces damping functions after all dilations, bounded thickenings, and translations needed by the Han–Schlag iteration. The damping-function FUP theorem then converts their fast decay into a single-scale observability estimate and iterates that estimate through the holes of $X$ .

Necessity of directional control

For $d\ge2$ , two orthogonal linear subspaces can be ball porous while Fourier duality maps the natural measure on one to the natural measure on the other. Line porosity excludes this model obstruction.

References

Alex Cohen, “Fractal uncertainty in higher dimensions,” 2024. arXiv record. Relevant: Theorem 1.1.