AI 中文总结
该研究分析线性散列的最大负载，证明其无多对数级最大负载，将界扩展到乘移散列族，建立其与算术 Kakeya 集密度变体的等价性，还表明改进其上界可推动算术 Kakeya 集研究。
AI 中文摘要
考虑经典的通用散列族 $h(x)=((ax+b)\text{ mod }p)\text{ mod }m$，其中 $a,b$ 从 $\mathbb Z_p$ 中均匀选取，我们将其称为线性散列，用于将 $n$ 个元素散列到 $m=\Theta(n)$ 个桶中。对于任何通用散列族，最大桶的期望大小至少为 $\Omega(\log n / \log\log n)$，至多为 $O(\sqrt{n})$。线性散列的这些平凡界的唯一改进是 Knudsen 在 2019 年给出的 $\tilde{O}(n^{1/3})$ 的上界。我们证明，对于任何足够大于 $n$ 的 $p$，存在一组 $n$ 个键，其期望最大负载为 $n^{\Omega(1/\log\log n)}$，从而证明线性散列没有多对数级的最大负载。我们将相同的界扩展到 Dietzfelbinger、Hagerup、Katajainen 和 Penttonen 提出的经典乘移散列族。我们证明了最大负载问题与算术 Kakeya 集的密度变体之间的等价性，随后使用 Green 和 Ruzsa 的构造完成下界，该构造包含一个小集合，其中包含指定范围内每个差值的长算术级数。令人惊讶的是，我们的等价性还意味着，任何对 Knudsen 上界的实质性改进都将暗示关于标准算术 Kakeya 集的新结果。
英文摘要
Consider the canonical universal hash family $h(x)= ((ax+b)\text{ mod } p)\text{ mod } m$, where $a,b$ are chosen uniformly from $\mathbb Z_p$, which we call linear hashing, being used to hash $n$ elements into $m=Θ(n)$ buckets. For any universal family, the expected size of the largest bucket is at least $Ω(\log n / \log\log n)$ and at most $O(\sqrt{n})$. The only improvement upon these trivial bounds for linear hashing is a 2019 upper bound of $\tilde{O}(n^{1/3})$ by Knudsen. We show that for any $p$ sufficiently larger than $n$, there is a set of $n$ keys whose expected maximum load is $n^{Ω(1/\log\log n)}$, proving linear hashing does not have a polylogarithmic maximum load. We extend the same bounds to the classical multiply-shift hash family of Dietzfelbinger, Hagerup, Katajainen, and Penttonen. Our main contribution is an equivalence between the maximum load problem to a density variant of arithmetic Kakeya sets. We then complete the lower bound using a construction of Green and Ruzsa of a small set containing long arithmetic progressions with every difference in a prescribed range. Surprisingly, our equivalence also implies that any substantial improvement over Knudsen's upper bound would imply new results about standard arithmetic Kakeya sets. More precisely, an $O(n^{1/3-\varepsilon})$ upper bound would improve known bounds for unions of complete integer arithmetic progressions, while an $n^{o(1)}$ upper bound would imply Bourgain's arithmetic-progression criterion.
