A generic threshold phenomena in weighted $\ell^2$

TL;DR

This work analyzes threshold phenomena in weighted $\ell^2$ spaces by establishing a summable Baire-category analogue of Körner's topological Ivashev-Musatov theorem and showing the optimality of the hypotheses. It develops a doubling-based framework and smooth localizers to construct bounded functions with controlled spectral data, and proves that generic small-support phenomena occur both for functions and for measures with $\ell^2$-Fourier coefficients. The results reveal sharp boundaries between continuity and singular behavior, and demonstrate that regularity conditions are essential through a strengthened Kantor–Meyer/Körner-type argument, with lacunary constructions illustrating limits of sparse Fourier-support techniques. Together, these results advance the understanding of uncertainty and threshold phenomena in harmonic analysis on the circle with weighted spectral norms.

Abstract

We consider threshold phenomenons in the context of weighted $\ell^2$-spaces. Our main result is a summable Baire category version of Körner's topological Ivashev-Musatov Theorem, which is proved to be optimal from several aspects.

Theorems & Definitions (20)

• Theorem 1.1
• Theorem 1.2
• Theorem 1.3
• Theorem 1.4
• Theorem 1.5
• Lemma 2.1
• proof
• Lemma 2.2
• Lemma 2.3: T. W. Körner
• proof : Proof of \ref{['LEM:KEYKÖR']}