离散傅里叶集中算子的迹亏界
Trace-defect bounds for discrete Fourier concentration operators
- The Graduate Center, City University of New York（纽约市立大学研究生院）
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中文总结 AI 辅助
本文研究离散傅里叶集中算子的迹亏，通过平移半范数给出其上下界，并应用于特征值计数与离散 Landau 渐近的恢复。
中文摘要 AI 辅助
设 $\Omega\subset\mathbb Z^d$ 为有限集，$S\subset\mathbb T^d$ 为可测集。我们研究离散傅里叶集中算子 \\[ T_{\Omega,S}=P_\Omega B_SP_\Omega, \\] 该算子描述了同时局域化到有限空间指标集和给定谱区域的过程。对于独立指数 $\gamma,\eta\in(0,1]$，我们引入 $\Omega$ 和 $S$ 的平移半范数，并利用它们来界定迹亏 \\[ \operatorname{tr}(T_{\Omega,S}-T_{\Omega,S}^2). \\] 证明结合了精确的迹亏恒等式与二进傅里叶估计，得到三种情形：当 $\gamma=\eta$ 时，各二进尺度的贡献同阶，其和产生对数因子；而当 $\gamma\ne\eta$ 时，该和由尺度范围的一端控制。对于满足指数为 $\gamma$ 的上 Minkowski 邻域估计的有界可测集 $F\subset\mathbb R^d$ 的离散化 $$ \Omega_R=(RF)\cap\mathbb Z^d $$，我们在临界情形下得到阶为 $R^{d-\gamma}\log R$ 的迹亏界，在临界线之外得到阶为 $R^{d-\min\{\gamma,\eta\}}$ 的迹亏界。这些估计给出了定量的特征值计数和 plunge 区域界，并恢复了离散 Landau 渐近。对于 $\gamma=\eta=1$ 的箱模型，对数因子是精确的。
英文摘要
Let $Ω\subset\mathbb Z^d$ 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 $γ,η\in(0,1]$, we introduce translation seminorms for $Ω$ 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 $γ=η$, the contributions from the dyadic scales are of the same order, and their sum produces a logarithmic factor, while for $γ\neη$ the sum is controlled by one end of the scale range. For discretizations $$ Ω_R=(RF)\cap\mathbb Z^d $$ of a bounded measurable set $F\subset\mathbb R^d$ satisfying an upper Minkowski-neighborhood estimate with exponent $γ$, we obtain trace-defect bounds of order $R^{d-γ}\log R$ 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.
