Trace-defect bounds for discrete Fourier concentration operators

RESUMO

Let $\Omega\subset\mathbb Z^d$ be finite and let $S\subset\mathbb T^d$ be measurable. We study the discrete Fourier concentration operator \[ T_{\Omega,S}=P_\Omega B_SP_\Omega, \] 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 \[ \operatorname{tr}(T_{\Omega,S}-T_{\Omega,S}^2). \] 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.