The Uncertainty Principle for certain densities

We prove a new version of the Uncertainty Principle of the form $\int |f|^2 \lesssim \int_{E^c} |f|^2 + \int_{\Sigma ^c}|\hat f|^2$ where the sets $E$ and $\Sigma$ are $\epsilon$-thin in the following sense: $|E \cap D(x, \rho_1(x))| \le \epsilon |D(x, \rho_1(x))|$ and $|\Sigma \cap D(x, \rho_2(x))| \le \epsilon |D(x, \rho_2(x))|$. This is an intermediate result between Logvinenko-Sereda's and Wolff's versions of the Uncertainty Principle.