The Uncertainty Principle for certain densities

Abstract

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.