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Asymmetric uniqueness sets in $\ell^q$

Adem Limani, Tomas Persson

Abstract

We exhibit an asymmetry phenomenon for uniqueness sets in $\ell^q$. Specifically, we construct sets that do not support measures with $\ell^q$-summable Fourier coefficients, yet simultaneously support measures whose positive frequencies decay faster than polynomials. In the language of Fourier uniqueness, this highlights a striking divergence between the unilateral and bilateral $\ell^q$ uniqueness problems.

Asymmetric uniqueness sets in $\ell^q$

Abstract

We exhibit an asymmetry phenomenon for uniqueness sets in . Specifically, we construct sets that do not support measures with -summable Fourier coefficients, yet simultaneously support measures whose positive frequencies decay faster than polynomials. In the language of Fourier uniqueness, this highlights a striking divergence between the unilateral and bilateral uniqueness problems.
Paper Structure (22 sections, 17 theorems, 108 equations)

This paper contains 22 sections, 17 theorems, 108 equations.

Table of Contents

  1. Introduction
  2. Unilateral versus bilateral Fourier decay
  3. Background
  4. Uniqueness sets and Fourier capacity
  5. Method and organization
  6. Entropy and unilateral Fourier decay
  7. Unilateral polynomial decay and Beurling--Carleson entropy
  8. Sets of uniqueness in $\ell^q$
  9. Characterizing uniqueness sets in $\ell^q$
  10. The main building blocks
  11. Sets of uniqueness for $\ell^q$ with entropic control
  12. Proof of \ref{['THM:MAIN']}
  13. Step 1. Extracting the main frequency masses
  14. Step 2. Disjoint frequency blocks with size control
  15. Step 3. Entropy control
  16. ...and 7 more sections

Key Result

Theorem 1.1

There exists a compact set $E\subset \mathbb{T}$ of Lebesgue measure arbitrarily close to $1$ such that:

Theorems & Definitions (28)

  • Theorem 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Corollary 1.4
  • Theorem 2.1: Khrushchev, 1973 khrushchev1978problem
  • Lemma 2.2
  • proof
  • Proposition 3.1
  • proof
  • proof : Proof of \ref{['THM:CHARlq']}
  • ...and 18 more