Dynamics Seminar

The fractal uncertainty principle (FUP) asserts that a function and its Fourier transform cannot both concentrate most of their $L^2$ mass near Cantor sets. Aside from being an interesting phenomenon of harmonic analysis, FUP implies that the Laplacian of convex cocompact hyperbolic manifolds has a spectral gap. I will discuss the proof of the case of FUP when the fractal is a subset of $\mathbb R^d$ and dimension $\leq d/2$, which I proved in joint work with James Leng and Zhongkai Tao. The result generalizes the $d = 1$ case due to Dyatlov and Jin, which in turn heavily used ideas of Dolgopyat and Naud for proving spectral gaps.