Probabilistic approach to FUP in the continuous setting for random Cantor sets in Ensembles I and II

Develop a probabilistic method to prove the fractal uncertainty principle for random Cantor sets constructed in Ensembles I and II in the continuous setting on R, despite the fact that the associated random Cantor measures may fail to have Fourier decay.

Background

The paper establishes strong Fourier decay and FUP for random Cantor sets constructed via Ensemble III in the continuous setting. However, the same Fourier decay approach may not work for random Cantor sets from Ensembles I and II because their measures can lack Fourier decay (even having Fourier dimension zero).

Consequently, extending the probabilistic FUP framework to these continuous ensembles remains open.