AI 中文总结
本文证明 Fourier 集中算子跃迁区特征值的单对数上界，通过正带限光滑化与几何递归分解，得到显式常数及波包相关估计，连接全局谱界与边界相消。
AI 中文摘要
我们证明了 Fourier 集中算子在跃迁区特征值的单对数界，在每一固定指数窗口内，对两个一般可容许集，去除了 Kulikov--Dam Larsen 界中的额外对数因子。对于有界空间集与频率集 $A,B\subset\mathbb{R}^d$，具有正测度且有限上余维一 Minkowski 边界含量，在 $(\varepsilon,1-\varepsilon)$ 中的特征值个数 $\Lambda_\varepsilon(cA,B)$ 满足 $\Lambda_\varepsilon(cA,B)\le C(d,A,B,\beta)c^{d-1}D(1+\log_+(c/D))$，其中 $D=\log(1/(\varepsilon(1-\varepsilon)))$，对每个固定 $\beta>0$，$c\ge2$，$0<\varepsilon<1/2$，且 $D\le\beta c$，这里 $\log_+t=\max(0,\log t)$。论证使用正带限光滑化，不需要对称性、凸性或连通性。在二维情形，相应的指数窗口估计具有对窗口参数一致的常数。我们还确定了每一维中球和标准单纯形的递归分解的最优坐标-Lipschitz 常数 $d-1$，并获得具有显式几何常数的盒频率估计。对于三维 $C^1$ 凸体，递归坐标斜率的最优下确界为 $2$。一致实阶 Bessel 估计在每个谱阈值给出独立的圆盘和球界，包括显式的深尾项。对于具有正曲率的光滑严格凸平面频率体，我们证明了任意相对尺度下对齐高斯波包的相关估计，以及分别估计空间块的代价的定量下界。这些结果共同将全局谱界与尖锐几何构造以及分组边界相互作用所保留的相消联系起来。
英文摘要
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 $A,B\subset\mathbb{R}^d$ of positive measure with finite upper codimension-one Minkowski boundary content, the number $Λ_\varepsilon(cA,B)$ of eigenvalues in $(\varepsilon,1-\varepsilon)$ satisfies $Λ_\varepsilon(cA,B)\le C(d,A,B,β)c^{d-1}D(1+\log_+(c/D))$, where $D=\log(1/(\varepsilon(1-\varepsilon)))$, for every fixed $β>0$, $c\ge2$, $0<\varepsilon<1/2$, and $D\leβc$, 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.
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