Trace-defect bounds for discrete Fourier concentration operators
Abstract
Let Ω⊂ℤd be finite and let S⊂𝕋d be measurable. We study the discrete Fourier concentration operator TΩ,S=PΩBSPΩ, which describes simultaneous localization to a finite set of spatial indices and a prescribed spectral region. For independent exponents γ,η∈(0,1], we introduce translation seminorms for Ω and S and use them to bound the trace defect tr(TΩ,S−TΩ,S2). The proof combines an exact trace-defect identity with dyadic Fourier estimates, leading to three regimes: when γ=η, the contributions from the dyadic scales are of the same order, and their sum produces a logarithmic factor, while for γ≠η the sum is controlled by one end of the scale range. For discretizations ΩR=(RF)∩ℤd of a bounded measurable set F⊂ℝd satisfying an upper Minkowski-neighborhood estimate with exponent γ, we obtain trace-defect bounds of order Rd−γlog R in the critical case and Rd−min{γ,η} off the critical line. These estimates yield quantitative eigenvalue-counting and plunge-region bounds and recover the discrete Landau asymptotic. For a box model at γ=η=1, the logarithmic factor is sharp.
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BibTeX
@misc{mayeli2026tracedefect,
title = {{Trace-defect bounds for discrete Fourier concentration operators}},
author = {Azita Mayeli},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.12226},
eprint = {2609.12226},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.12226},
}
RIS
TY - UNPB
TI - Trace-defect bounds for discrete Fourier concentration operators
AU - Azita Mayeli
PY - 2026
DA - 2026/09/10
PB - arXiv
DO - 10.48550/arxiv.2609.12226
UR - https://arxiv.org/abs/2609.12226
AB - Let Ω⊂ℤ^d be finite and let S⊂𝕋^d be measurable. We study the discrete Fourier concentration operator T_Ω,S=P_ΩB_SP_Ω, which describes simultaneous localization to a finite set of spatial indices and a prescribed spectral region. For independent exponents γ,η∈(0,1], we introduce translation seminorms for Ω and S and use them to bound the trace defect tr(T_Ω,S−T_Ω,S^2). The proof combines an exact trace-defect identity with dyadic Fourier estimates, leading to three regimes: when γ=η, the contributions from the dyadic scales are of the same order, and their sum produces a logarithmic factor, while for γ≠η the sum is controlled by one end of the scale range. For discretizations Ω_R=(RF)∩ℤ^d of a bounded measurable set F⊂ℝ^d satisfying an upper Minkowski-neighborhood estimate with exponent γ, we obtain trace-defect bounds of order R^d−γlog R in the critical case and R^d−min{γ,η} off the critical line. These estimates yield quantitative eigenvalue-counting and plunge-region bounds and recover the discrete Landau asymptotic. For a box model at γ=η=1, the logarithmic factor is sharp.
ER -