An independent proof of the plunge-region conjecture for time-frequency localization operators in dimension one

Abstract

Let $A_0,B_0\subset\mathbb{R}$ be bounded measurable sets of positive measure with finite topological boundaries, and let $S_{cA_0,B_0}=P_{cA_0}Q_{B_0}P_{cA_0}$ be the associated time-frequency localization operator, where $P_E$ is multiplication by $\mathbf{1}_E$ and $Q_E=\mathcal{F}^{-1}P_E\mathcal{F}$. We prove that the plunge count $Λ_\varepsilon=\#\{n:\varepsilon<λ_n(S_{cA_0,B_0})<1-\varepsilon\}$ satisfies $Λ_\varepsilon \le C(A_0,B_0)\,\widetilde{L}\,(1+\ln_+(ca/\widetilde{L}))$, with $\widetilde{L}=\ln(1/(\varepsilon(1-\varepsilon)))$, for all $c>0$ and $0<\varepsilon<1/2$, where $a$ is the largest component length of $A_0$ and $C(A_0,B_0)$ is explicit. In particular this establishes, in dimension $d=1$, the conjecture of Kulikov and Dam Larsen (arXiv:2603.23832). The proof does not invoke the Kulikov-Dam Larsen parallelepiped theorem or any prolate-spheroidal or Chebyshev-polynomial spectral machinery for $S$ itself. Instead it works directly with the off-diagonal factor $T=P_{(cA_0)^c}Q_{B_0}P_{cA_0}$: an exact oscillation factorization special to $d=1$ reduces each one-sided, one-scale piece of $T$ to a fixed Hankel kernel $1/(2π(s+r))$; a scale-uniform Bernstein-ellipse estimate gives geometric singular-value decay; and a Taylor-rank bound controls the boundary layer. The pieces are assembled by the Rotfel'd $p$-quasi-norm inequality over $O(\log)$ dyadic scales of a one-variable decomposition, sidestepping the failure of Cotlar-Stein almost-orthogonality in Schatten $p$-quasi-norms. We indicate precisely which steps are specific to $d=1$.

Disclosure

“d nothing proved above settles it. Of the three d = 1-specific ingredients, (b) is the obstruction we cannot presently circumvent; whether (a) and (c) admit substitutes in d ≥ 2 we leave open. Acknowledgments AI Use Disclosure. Generative AI tools were used to assist with language editing and copy- editing of this manuscript. Selected equations and derivations were additionally checked using an internal model-verification pipeline developed by Harmonic Research Technologies (H”

PDF page 14

Classification

Rewriting existing author-written text

Multiplier

4

Verified

2026-08-23T19:06:15.372575Z

Structural counts