Damping-function fractal uncertainty theorem

Statement

Let $X\subseteq[-1,1]^d$ be $\nu$ -porous on balls from scales $h$ to $1$ , and let $Y\subseteq[-h^{-1},h^{-1}]^d$ . Assume there are fixed $c_2,c_3,\alpha\in(0,1)$ such that, for every $h<s<1$ and every allowed translation $\eta$ ,

sY+[-4,4]^d+\eta

admits a damping function with spectral radius $c_1=20\nu\sqrt d$ and parameters $c_2,c_3,\alpha$ . Then constants $C,\beta>0$ , depending only on these parameters and $d$ , satisfy

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

Iteration mechanism

Damping functions yield a uniform quantitative unique-continuation estimate. It shows that at each scale a fixed portion of $f$ 's mass lies in the holes of $X$ . Iterating over order $\log(h^{-1})$ scales turns the fixed loss into the power $h^\beta$ .

Convention

Equivalent versions use $\ell^1$ cubes and $\ell^1$ -decay. Norm equivalence in finite-dimensional Euclidean space changes the constants but not the theorem.

References

Rui Han and Wilhelm Schlag, “A higher-dimensional Bourgain–Dyatlov fractal uncertainty principle,” Analysis & PDE 13 (2020), Theorem 5.1. DOI record.
Alex Cohen, “Fractal uncertainty in higher dimensions,” 2024. arXiv record. Relevant: Theorem 1.6 and Appendix A.1.