The motivation of the question comes from the paper "Some harmonic analysis questions suggested by Anderson-Bernoulli models. Geom. Funct. Anal. 8 (1998), no. 5, 932–964" by Shubin, Vakilian and Wolff. In this paper, the authors first prove a version of the uncertainty inequality for $L^2(\mathbb{R}^n) $ functions. More precisely, they prove that there exist $\epsilon>0$ and $C>0$ such that $$ \|f\|_2\leq C\left(\|f\|_{L^2(E^c)}+\|\hat{f}\|_{L^2(F^c)}\right) $$ hold when $E, F$ are $\epsilon-$ thin sets. Using this inequality they give an estimate for the norm of the product of two operators which assign functions with the Fourier transforms of its function multiples when the multipliers are dominated by certain probability measures (Theorem 2.3). These results are applied to one-dimensional Anderson models and Bernoulli models.

I'm not familiar with the problem of Andersion localization, I only know that this phenomenon named after Anderson, who suggested a mechanism for electron localization is in a lattice potential, provided that the degree of randomness (disorder) in the lattice is suciently large.

I'm quite surprised to see that uncertainty princple can be applied in this problem. Can someone explain a little bit on the relationship between the two principles or provide some intuition that why the Uncertainty Princple works in Andersion localization? I'm also apprreciated that more examples can be given to understand the relation better.