双侧线性哈希与光滑格覆盖的二次密度界
Two-sided linear hashing and quadratic density bounds for smooth lattice coverings
- Xidian University（西安电子科技大学）
- Institute of Mathematics and Interdisciplinary Studies（数学与交叉科学研究院）
机构由 AI 辅助整理，请以论文原文为准。
中文总结 AI 辅助
本研究提出双侧线性哈希方法，通过随机线性投影的纤维大小界，将光滑格覆盖的复杂度从 O(n^3) 改进至 O(n^2)，并证明该阶数最优。
中文摘要 AI 辅助
我们研究有限域子集的随机线性投影，其中每个纤维的基数都接近其平均值。我们限定了确保所有纤维满足给定相对偏差（以给定失败概率）所需的平均纤维大小。对于投影到 $\mathbb F_q^b$ 的 $S\subseteq\mathbb F_q^n$，一个定理给出了三种情形：在固定偏差和失败概率下，足够的平均纤维大小对于任意 $q$ 为 $O(q2^b)$，当 $q$ 至少是 $b$ 的适当常数倍时为 $O(q^2)$，对于固定 $q$ 为 $O_q(b)$。在固定域上产生的熵损失为 $h-b=\log_q h+O(1)$，其中 $h=\log_q|S|$ 是输入熵。这与 Alon、Dietzfelbinger、Miltersen、Petrank 和 Tardos（1999）的二元障碍的阶数相匹配；我们给出了每个固定域上的定量随机源细化。我们的证明将商与平均计数引理与 Dhar 和 Dvir（arXiv:2204.01665）的局部平衡/非平衡论证以及 Dhar、Dvir 和 Kumar 和 Mon（arXiv:2609.17020）的 Furstenberg 估计相结合。我们将这些界应用于 Ordentlich、Regev 和 Weiss（arXiv:2311.04644）的归约中，将其光滑格覆盖的 $O(n^3)$ 界改进为 $O(n^2)$。对于每个固定凸体 $K\subseteq\mathbb R^n$，协体积为一的 Haar-Siegel 随机格在 $K$ 的每个平移中的格点数与 $\operatorname{vol}(K)$ 的相对误差在给定范围内，且具有给定的高概率，只要 $\operatorname{vol}(K)\ge Cn^2$ 且 $n$ 足够大。常数和维度截止仅取决于误差和失败概率。Kopparty、Lev、Saraf 和 Sudan（arXiv:1003.3736）的高秩 Kakeya 集的补集表明，在相同归约中，任意子集的哈希保证无法产生更小的阶数。
英文摘要
We study random linear projections of a finite-field subset for which every fiber has cardinality close to its mean. We bound the mean fiber size needed to ensure that all fibers satisfy a prescribed relative discrepancy, with a prescribed failure probability. For $S\subseteq\mathbb F_q^n$ projected to $\mathbb F_q^b$, one theorem gives three regimes: at fixed discrepancy and failure probability, sufficient mean fiber sizes are $O(q2^b)$ for arbitrary $q$, $O(q^2)$ when $q$ is at least a suitable constant multiple of $b$, and $O_q(b)$ for fixed $q$. The resulting entropy loss over fixed fields is $h-b=\log_q h+O(1)$, where $h=\log_q|S|$ is the input entropy. This matches the order of the binary obstruction of Alon, Dietzfelbinger, Miltersen, Petrank, and Tardos (1999); we give a quantitative random-source refinement over every fixed field. Our proof combines a quotient-and-average counting lemma with the local balanced/unbalanced argument of Dhar and Dvir (arXiv:2204.01665) and Furstenberg estimates of Dhar and Dvir and Kumar and Mon (arXiv:2609.17020). We apply these bounds in the reduction of Ordentlich, Regev, and Weiss (arXiv:2311.04644) to improve their $O(n^3)$ bound for smooth lattice coverings to $O(n^2)$. For each fixed convex body $K\subseteq\mathbb R^n$, a Haar-Siegel random lattice of covolume one has the number of lattice points in every translate of $K$ within a prescribed relative error of $\operatorname{vol}(K)$, with prescribed high probability, once $\operatorname{vol}(K)\ge Cn^2$ and $n$ is sufficiently large. The constant and dimension cutoff depend only on the error and failure probability. Complements of higher-rank Kakeya sets of Kopparty, Lev, Saraf, and Sudan (arXiv:1003.3736) show that no hashing guarantee for arbitrary subsets can yield a smaller order in the same reduction.
