A Logvinenko-Sereda theorem for lacunary spectra

For a function $F$ represented as $F(x)=\sum_{n=0}^\infty{f_n (x) e^{2 \pi i \lambda_n x}},$ where each $f_n$ satisfies $\operatorname{spec}(f_n) \subset [0, 1]$ and $(\lambda_n)_{n\geq 0}\subset \mathbb{R}_+$ is a lacunary sequence, we obtain $$\|F\|_{L^2(\mathbb{R})}\lesssim \|F\chi_{E}\|_{L^2(\mathbb{R})}$$ 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.