一类随机对象的计数函数
Counting Functions for Random Objects in a Category
摘要：
在算术统计和解析数论中，给出阶数低于$X$的对象数的计数函数的渐近增长率被研究为$X\to \infty$。我们定义了一般的计数函数，它可以在某种顺序下对某个范畴上的对象的满态进行计数。给定一个概率测度$μ$的同构类的范畴与足够的尊重的产品结构，我们证明了一个版本的大数定律，以给出渐近增长率$X$趋于$\infty$的概率$1$的功能，在有限时刻的$μ$和订购。这样的计数函数是由算术统计学中的工作所激发的，包括马勒猜想中的数域计数和巴季列夫-马宁猜想中的点计数。最近的工作Sawin-木材给出了充分条件来构建这样一个措施$μ$从一个行为良好的序列有限时刻在非常广泛的背景下，我们证明了我们的结果在这个广泛的背景下，增加了假设，产品结构的类别是尊重。这些结果使我们能够在一般设置中形式化关于计数函数的大量启发式预测。
英文摘要：
In arithmetic statistics and analytic number theory, the asymptotic growth rate of counting functions giving the number of objects with order below $X$ is studied as $X\to \infty$. We define general counting functions which count epimorphisms out of an object on a category under some ordering. Given a probability measure $μ$ on the isomorphism classes of the category with sufficient respect for a product structure, we prove a version of the Law of Large Numbers to give the asymptotic growth rate as $X$ tends towards $\infty$ of such functions with probability $1$ in terms of the finite moments of $μ$ and the ordering. Such counting functions are motivated by work in arithmetic statistics, including number field counting as in Malle's conjecture and point counting as in the Batyrev-Manin conjecture. Recent work of Sawin--Wood gives sufficient conditions to construct such a measure $μ$ from a well-behaved sequence of finite moments in very broad contexts, and we prove our results in this broad context with the added assumption that a product structure in the category is respected. These results allow us to formalize vast heuristic predictions about counting functions in general settings.
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