Fractal uncertainty principle via Dolgopyat’s method in higher dimensions

The HADES seminar on Tuesday, March 21st will be at 3:30 pm in Room 740.

Speaker: Zhongkai Tao

Abstract: The fractal uncertainty principle (FUP) was introduced by Dyatlov–Zahl which states that a function cannot be localized near a fractal set in both position and frequency spaces. It has rich applications in proving spectral gaps and quantum chaos. Bourgain–Dyatlov proved the fractal uncertainty principle in dimension $1$, which leads to an essential spectral gap, and was applied by Dyatlov–Jin and Dyatlov–Jin–Nonnenmacher to show quantum limits on closed negatively curved surfaces have full support. The higher dimensional version of the fractal uncertainty principle for large fractal sets is widely open, and there is a recent work by Alex Cohen who addressed the case of $2$ dimensional arithmetic Cantor sets.

I will talk about the history of the fractal uncertainty principle and explain its applications via examples. Then I will talk about our recent work, joint with Aidan Backus and James Leng, which proves a fractal uncertainty principle for small fractal sets, improving the volume bound in higher dimensions. This generalizes the work of Dyatlov–Jin using Dolgopyat’s method. As an application, we get effective essential spectral gaps for convex cocompact hyperbolic manifolds in higher dimensions with Zariski dense fundamental groups. The new ingredients include a “non-orthogonality condition”, an explicit construction of Christ cubes and a statistical argument.

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