Trace-defect bounds for discrete Fourier concentration operators

Abstract

Let $\Omega \subset {\mathbb{Z}}^d$ be finite and let $S\subset {\mathbb{T}}^d$ 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 $\gamma ,\eta \in (0,1]$, we introduce translation seminorms for $\Omega$ and $S$ and use them to bound the trace defect \[ \operatorname{tr}(T_{Ω,S}-T_{Ω,S}^2). \] The proof combines an exact trace-defect identity with dyadic Fourier estimates, leading to three regimes: when $\gamma =\eta$, the contributions from the dyadic scales are of the same order, and their sum produces a logarithmic factor, while for $\gamma \ne \eta$ the sum is controlled by one end of the scale range. For discretizations $${\Omega }_R=(RF)\cap {\mathbb{Z}}^d$$ of a bounded measurable set $F\subset {\mathbb{R}}^d$ satisfying an upper Minkowski-neighborhood estimate with exponent $\gamma$, we obtain trace-defect bounds of order $R^{d-\gamma }\log R$ in the critical case and $R^{d-\min \{\gamma ,\eta \}}$ 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 $\gamma =\eta =1$, the logarithmic factor is sharp.

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