A Logvinenko-Sereda theorem for lacunary spectra

TL;DR

This paper extends the Logvinenko-Sereda theorem to functions with lacunary spectra, showing that their global L^2 norm can be controlled by their norm on thick subsets, answering a question for functions with positive frequencies.

Contribution

It generalizes the Logvinenko-Sereda theorem to lacunary spectral functions, providing new bounds for functions with positive frequencies.

Findings

L^2 norm of lacunary spectrum functions is controlled by thick subsets.

Extension of Logvinenko-Sereda theorem to positive frequency functions.

Addresses a question posed by Kovrizhkin.

Abstract

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.