A Logvinenko-Sereda theorem for lacunary spectra

For a function $F$ represented as $F(x)={\sum }_{n=0}^{\mathrm{\infty }}{f}_n(x)e^{2\pi i{\lambda }_{n}x},$ where each $f_n$ satisfies $\mathrm{spec}(f_n)\subset [0,1]$ and $({\lambda }_n{)}_{n\ge 0}\subset {\mathbb{R}}_{+}$ is a lacunary sequence, we obtain $$\mathrm{\parallel }F{\mathrm{\parallel }}_{L^2(\mathbb{R})}\lesssim \mathrm{\parallel }F{\chi }_E{\mathrm{\parallel }}_{L^2(\mathbb{R})}$$ provided that $E$ is a thick subset of $\mathbb{R}$. This extends the Logvinenko-Sereda theorem and answers a question posed by Kovrizhkin for functions with positive frequencies.