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A Logvinenko-Sereda theorem for lacunary spectra

Miquel Saucedo, Sergey Tikhonov

Abstract

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

A Logvinenko-Sereda theorem for lacunary spectra

Abstract

For a function represented as where each satisfies and is a lacunary sequence, we obtain provided that is a thick subset of . This extends the Logvinenko-Sereda theorem and answers a question posed by Kovrizhkin for functions with positive frequencies.
Paper Structure (7 sections, 9 theorems, 65 equations)

This paper contains 7 sections, 9 theorems, 65 equations.

Table of Contents

  1. Introduction
  2. Main result
  3. Structure of the paper
  4. Proof of Theorem \ref{['theorem:mainth']}
  5. Further remarks
  6. Sets of uniqueness for lacunary spectra
  7. Some remarks about strong lacunary sequences

Key Result

Theorem 1.1

Let $F^c=[0,1]$. Then inequality ls holds for a measurable set $E$ if and only if $E$ is ($\Delta,\gamma$)-thick (or, simply, thick), that is, if and only if there exist $\Delta,\gamma>0$ such that for every interval $I$ of length $\Delta$, Moreover, one can take $C:=C(\gamma,\Delta)$ in inequality ls.

Theorems & Definitions (16)

  • Theorem 1.1
  • Definition 1.3: Strong Zygmund lacunary
  • Theorem 1.4
  • Lemma 2.1
  • Theorem 2.2
  • proof : Proof of Lemma \ref{['lemmamain']}
  • Proposition 2.3
  • proof
  • Lemma 2.4
  • proof : Proof of Theorem \ref{['theorem:mainth']}
  • ...and 6 more