AI 中文总结
本文改进了数字段上整数点的更大筛法界，并应用二维筛法获得Bombieri-Pila类型的界。
AI 中文摘要
设A是数域K上的代数整数集合，其高度不超过H，并且对于每个素理想p，满足|A mod p|≤α|O_K/p|，其中Np>c，α∈(0,1)。根据Ellenberg、Elsholtz、Hall和Kowalski等人提出的更大筛法，可得|A|≪_{K,c,α}H^{2α}。本文改进了这一更大筛法的界，证明|A|≪_{K,c,α}H^α(log H)^r。我们还获得了Helfgott和Venkatesh类型的二维更大筛法，应用于数域O_K上的Bombieri-Pila类型界。
英文摘要
Let $A \subseteq \mathcal{O}_{K}$ be a set of algebraic integers of height up to $H$ such that $|A \mod{\mathfrak{p}}|\leq α|\mathcal{O}_{K}/\mathfrak{p}|$ for every prime ideal $\mathfrak{p}$ with $N\mathfrak{p}>c$ for some $α\in (0,1)$. It follows from a larger sieve due to Ellenberg, Elsholtz, Hall and Kowalski that $|A| \ll_{K,c,α}H^{2α}$. In this paper, we improve on this larger sieve bound by showing that $|A|\ll_{K,c,α}H^α(\log H)^r$. We also obtain a two-dimensional larger sieve of Helfgott and Venkatesh type over $\mathcal{O}_{K} \times \mathcal{O}_{K}$ and apply it to produce a Bombieri-Pila type bound over $\mathcal{O}_{K}$.
