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Generic measures with slowly decaying Fourier coefficients

Adem Limani

Abstract

We investigate threshold phenomena in weighted $\ell^2$-spaces and characterize the critical regimes where elements with either small support or maximally bad range can be constructed. Our results are shown to be optimal in several respects, and our proofs principally rely on techniques involving sparse Fourier spectrum. We further show that these seemingly pathological constructions are actually generic from certain categorical perspectives.

Generic measures with slowly decaying Fourier coefficients

Abstract

We investigate threshold phenomena in weighted -spaces and characterize the critical regimes where elements with either small support or maximally bad range can be constructed. Our results are shown to be optimal in several respects, and our proofs principally rely on techniques involving sparse Fourier spectrum. We further show that these seemingly pathological constructions are actually generic from certain categorical perspectives.

Paper Structure

This paper contains 20 sections, 19 theorems, 123 equations.

Table of Contents

  1. Introduction
  2. Supports of elements in weighted $\ell^2$
  3. Purpose and aim
  4. Main results
  5. Positive measures with small support
  6. Functions with topologically wild range
  7. Organization and notation.
  8. Acknowledgments
  9. Functions with wild range
  10. Riesz products
  11. An adapted family of Riesz products
  12. The construction of elements with maximally bad range
  13. Functional analytic setting and duality
  14. A Baire category argument
  15. Measures with small support
  16. ...and 5 more sections

Key Result

Theorem 1.1

Let $(\lambda_n)_n$ be positive numbers satisfying the following conditions: Then there exists a positive function $f\in L^\infty(\mathbb{T},dm)$ such that $\text{supp}({fdm})$ contains no interior, and

Theorems & Definitions (34)

  • Theorem 1.1: See limani2025generic
  • Theorem 1.2
  • Theorem 2.1
  • Theorem 2.2
  • Proposition 2.3
  • Theorem 2.4
  • Theorem 2.5
  • Lemma 3.1
  • proof
  • Lemma 3.2
  • ...and 24 more