A Logvinenko-Sereda theorem for lacunary spectra

Abstract

For a function $F$ represented as $F(x)={\sum }_{n=0}^{\infty }{f}_n(x)e^{2\ddot{I}€i\hat{I}{»}_{n}x},$ where each $f_n$ satisfies $\mathrm{spec}(f_n)\subset [0,1]$ and $(\hat{I}{»}_n{)}_{n\ge 0}\subset {\mathbb{R}}_{+}$ is a lacunary sequence, we obtain $$\Vert F{\Vert }_{L^2(\mathbb{R})}\lesssim \Vert F\ddot{I}{‡}_E{\Vert }_{L^2(\mathbb{R})}$$ 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.