The Uncertainty Principle for certain densities

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We prove a new version of the Uncertainty Principle of the form ∫ |f|² ∫ₑᶜ |f|² + ∫Σᶜ| f|² where the sets E and Σ are ε-thin in the following sense: |E ∩ D(x, ρ₁(x))| ≤ ε|D(x, ρ₁(x))| and |Σ∩ D(x, ρ₂(x))| ≤ ε|D(x, ρ₂(x))|. This is an intermediate result between Logvinenko-Sereda's and Wolff's versions of the Uncertainty Principle.