任意凸体的近线性格覆盖
Nearly Sharp Bounds for Lattice Coverings by Convex Bodies
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中文总结 AI 辅助
该研究针对n维凸体的格覆盖密度，通过垂直-水平构造等方法，将其最坏情形增长指数优化为近线性，改进了此前O(n²)的通用界，确定了最优多项式增长指数。
中文摘要 AI 辅助
对于n维凸体K，记其格覆盖密度为θ_L(K)。我们证明存在绝对常数C>0，使得对所有这类凸体一致满足θ_L(K)≤Cn log n (log log n)^(10/3+o(1))，其中o(1)项与K无关。该结果改进了Ordentlich、Regev和Weiss之前的通用界O(n²)，并结合Coxeter、Few与Rogers针对欧氏球的经典线性下界，确定了最坏情形格覆盖密度的最优多项式增长指数。核心创新是一种垂直-水平构造，将全布尔立方体上的均匀权重估计转化为任意凸体的格覆盖：在低维垂直空间中，折叠高斯函数与有限域Kakeya定理为每个平移选择总归一化权重为d^(-5/2+o(1))的布尔模式；高斯边际将所得权重转移至凸体的截面，熵方法则生成一个未覆盖集极小的水平格；有限指标提升将所有平移组装为单个格，再通过Rogers完备引理实现精确覆盖。
英文摘要
For an $n$-dimensional convex body $K$, let $θ_L(K)$ denote its lattice covering density, and let $Θ_L^{\mathrm{conv}}(n)$ and $Θ_L^{\mathrm{sym}}(n)$ be the corresponding worst-case quantities over all convex bodies and over origin-symmetric convex bodies, respectively. Before this work, these quantities were known only to lie between a lower bound of order $n$ and an upper bound of order $n^2$, so even their polynomial order was undetermined. We prove that there are absolute constants $c,C>0$ such that \[ c n\log n \le Θ_L^{\mathrm{sym}}(n) \le Θ_L^{\mathrm{conv}}(n) \le Cn\log n\,(\log\log n)^{10/3+o(1)}. \] Thus both worst-case quantities are $n\log n\,(\log n)^{o(1)}$, and the upper and lower bounds differ by a factor at most $(\log\log n)^{10/3+o(1)}$. For the upper bound, a vertical--horizontal amplification based on weighted Boolean cubes combines covering estimates for low-codimensional sections into an exact lattice covering of an arbitrary convex body. For the lower bound, a random-slab construction and Poisson witnesses on flat tori show, with positive probability, that the resulting body admits no lattice covering of density below $c n\log n$.
