The HADES seminar on Tuesday, October 18th will be at 3:30 pm over zoom. Zoom link:https://berkeley.zoom.us/j/96232331895.

Speaker: Alex Cohen

Abstract: A fractal uncertainty principle (FUP) states that a function $f$ and its Fourier transform cannot both be large on a fractal set. These were recently introduced by Semyon Dyatlov and collaborators in order to prove new results in quantum chaos. So far FUPs are only understood for fractal sets in $\mathbb{R}$, and fractal sets in $\mathbb{R}^2$ remain elusive. In this talk, we prove a sharp fractal uncertainty principle for Cantor sets in $\mathbb{Z}/N\mathbb{Z} \times \mathbb{Z}/N\mathbb{Z}$, a discrete model for $\mathbb{R}^2$. The main tool is a quantitative form of Lang’s conjecture from number theory due to Beukers and Smyth.