Fractal uncertainty principle

Definition

A fractal uncertainty principle is a quantitative estimate asserting that a function cannot concentrate simultaneously near a fractal set $X$ in physical space and a fractal set $Y$ in Fourier space. In the scaled Euclidean form used here, it is an implication

\operatorname{supp}\widehat f\subseteq Y \quad\Longrightarrow\quad \|f\mathbf1_X\|_{L^2}\le C h^\beta\|f\|_{L^2},

where $0<h\ll1$ , $C<\infty$ , and the gain exponent $\beta>0$ are uniform over sets satisfying the stated multiscale hypotheses.

Why the power saving matters

An estimate obtained only from the volumes of $X$ and $Y$ is called the trivial uncertainty bound. An FUP supplies an additional positive power of $h$ , or remains nontrivial in a regime where the volume bound gives no decay. Its force comes from holes occurring at many scales rather than from small measure at just one scale.

Higher-dimensional geometry

In dimensions at least two, ball porosity of both sets is insufficient because physical and Fourier concentration can live on orthogonal subspaces. The line-porous FUP rules out that obstruction by requiring line porosity on the Fourier-side set.

Quantum-chaos role

Fractal sets arise from trajectories that avoid an observation region in a chaotic flow. The FUP turns their geometric holes into observability, eigenfunction mass bounds, or resonance gaps.

References

Semyon Dyatlov, “An introduction to fractal uncertainty principle,” Journal of Mathematical Physics 60 (2019), 081505. DOI record.
Jean Bourgain and Semyon Dyatlov, “Spectral gaps without the pressure condition,” Annals of Mathematics 187 (2018), 825–867. DOI record.