Caltech/UCLA Joint Analysis Seminar

Recently Bourgain and Dyatlov proved a Fractal Uncertainty Principle (FUP) which roughly says that: Assuming a function on R R has its Fourier support contained in a fractal set Y Y. Then its L 2 L2 norm on a fractal set X X, whose scale is dual to that of Y Y, cannot be close to its L 2 L2 norm over the whole R R as long as dimX,dimY<1 dim⁡X,dim⁡Y<1. The proof seems quite interesting and unusual to me. I will talk about the ingredients in the proof, including the Beurling-Malliavin Theorem. In the original work of Bourgain and Dyatlov the FUP was ineffective. I will also talk about why we can obtain an effective version (joint work with Long Jin) by looking closely into the proof of the Beurling-Malliavin Theorem and proving a weaker but "more effective" version of it.