Plunge bounds for Fourier concentration operators: geometry, smoothing, and wave packets
Abstract
We prove a single-logarithm bound for the eigenvalues in the plunge region of Fourier concentration operators, removing the extra logarithmic factor in the Kulikov--Dam Larsen bound for two general admissible sets throughout each fixed exponential window. For bounded spatial and frequency sets A,B⊂ℝd of positive measure with finite upper codimension-one Minkowski boundary content, the number Λε(cA,B) of eigenvalues in (ε,1−ε) satisfies Λε(cA,B)≤ C(d,A,B,β)cd−1D(1+log+(c/D)), where D=log(1/(ε(1−ε))), for every fixed β>0, c≥2, 0<ε<1/2, and D≤βc, with log+t=max(0,log t). The argument uses positive band-limited smoothing and requires no symmetry, convexity, or connectedness. In dimension two, the corresponding exponential-window estimate has a constant uniform in the window parameter. We also determine the optimal coordinate-Lipschitz constant d−1 for recursive decompositions of balls and standard simplices in every dimension, and obtain box-frequency estimates with explicit geometric constants. For C1 convex bodies in three dimensions, the optimal infimum of recursive coordinate slopes is $2$. Uniform real-order Bessel estimates give independent disk and ball bounds at every spectral threshold, including an explicit deep-tail term. For smooth strictly convex planar frequency bodies with positive curvature, we prove correlation estimates for aligned Gaussian packets at arbitrary relative scales and quantitative lower bounds on the cost of estimating spatial blocks separately. Together these results connect the global spectral bound with sharp geometric constructions and the cancellation retained by grouping boundary interactions.
Related
- Trace-defect bounds for discrete Fourier concentration operators
- Local Smoothing Estimates of Schrödinger Operators on Waveguides Using Refined Decoupling
- The Geometry of Randomized Smoothing on Feasible Sets
- Fourier decay and endpoint estimates for lacunary maximal operators
- The concentration conjecture for Dirac--Coulomb operators
BibTeX
@misc{azimifard2026plunge,
title = {{Plunge bounds for Fourier concentration operators: geometry, smoothing, and wave packets}},
author = {Ahmadreza Azimifard},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.09406},
eprint = {2610.09406},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.09406},
}
RIS
TY - UNPB
TI - Plunge bounds for Fourier concentration operators: geometry, smoothing, and wave packets
AU - Ahmadreza Azimifard
PY - 2026
DA - 2026/10/07
PB - arXiv
DO - 10.48550/arxiv.2610.09406
UR - https://arxiv.org/abs/2610.09406
AB - We prove a single-logarithm bound for the eigenvalues in the plunge region of Fourier concentration operators, removing the extra logarithmic factor in the Kulikov--Dam Larsen bound for two general admissible sets throughout each fixed exponential window. For bounded spatial and frequency sets A,B⊂ℝ^d of positive measure with finite upper codimension-one Minkowski boundary content, the number Λ_ε(cA,B) of eigenvalues in (ε,1−ε) satisfies Λ_ε(cA,B)≤ C(d,A,B,β)c^d−1D(1+log_+(c/D)), where D=log(1/(ε(1−ε))), for every fixed β>0, c≥2, 0<ε<1/2, and D≤βc, with log_+t=max(0,log t). The argument uses positive band-limited smoothing and requires no symmetry, convexity, or connectedness. In dimension two, the corresponding exponential-window estimate has a constant uniform in the window parameter. We also determine the optimal coordinate-Lipschitz constant d−1 for recursive decompositions of balls and standard simplices in every dimension, and obtain box-frequency estimates with explicit geometric constants. For C^1 convex bodies in three dimensions, the optimal infimum of recursive coordinate slopes is $2$. Uniform real-order Bessel estimates give independent disk and ball bounds at every spectral threshold, including an explicit deep-tail term. For smooth strictly convex planar frequency bodies with positive curvature, we prove correlation estimates for aligned Gaussian packets at arbitrary relative scales and quantitative lower bounds on the cost of estimating spatial blocks separately. Together these results connect the global spectral bound with sharp geometric constructions and the cancellation retained by grouping boundary interactions.
ER -