The Fourier uncertainty principle describes a fundamental phenomenon that a function and its Fourier transform cannot simultaneously localize. Dyatlov and his collaborators recently introduced a concept of Fractal Uncertainty Principle (FUP). It is a mathematical formulation concerning the limit of localization of a function and its Fourier transform on sets with certain fractal structure.

 

The FUP has quickly become an emerging topic in Fourier analysis and also has important applications to other fields such as quantum chaos. In this talk, we report on an ongoing project concerning the FUP when the fractal sets are constructed via certain random procedures. Examples include random Cantor sets in the discrete or continuous setting. We present the FUP with a much more favorable estimate than the ones in the deterministic cases. We also propose questions and applications of the FUP by this probabilistic approach. The talk is based on joint works with Suresh Eswarathasan and Pouria Salekani.