2026/07/25 by Ahmadreza Azimifard · 1 citation
Mathematics · #math.FA #math.CA #math.SP
Let A0,B0⊂ℝ be bounded measurable sets of positive measure with finite topological boundaries, and let ScA0,B0=PcA0QB0PcA0 be the associated time-frequency localization operator, where PE is multiplication by 1E and QE=F-1PEF. We prove that the plunge count Λε=#\n:ε<λn(ScA0,B0)<1-ε\ satisfies Λε ≤ C(A0,B0) \widetildeL (1+ln+(ca/\widetildeL)), with \widetildeL=ln(1/(ε(1-ε))), for all c>0 and 0<ε<1/2, where a is the largest component length of A0 and C(A0,B0) 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(cA0)cQB0PcA0: 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.