I will talk about the fractal uncertainty principle in the regime of the volume bound. The interest in such estimates comes from potential applications to spectral gaps in dynamical systems at high frequencies. I will explain the proof in the Ahlfors--David regular case, which was done by Dyatlov and Jin in dimension one, and by Backus, Leng, and myself in higher dimensions under a non-orthogonality condition. I will also explain the relationship of this estimate with recent Fourier decay estimates.
This talk is based on joint work with Aidan Backus and James Leng.
