Larger sieve with height function and uniform bounds for integral points on curves over number fields

Abstract

Let $A\subseteq {\mathcal{O}}_K$ be a set of algebraic integers of height up to $H$ such that $|A\mathrm{mod}\mathfrak{p}|\le \hat{I}\pm |{\mathcal{O}}_K/\mathfrak{p}|$ for every prime ideal $\mathfrak{p}$ with $N\mathfrak{p}>c$ for some $\hat{I}\pm \in (0,1)$. It follows from a larger sieve due to Ellenberg, Elsholtz, Hall and Kowalski that $|A|{\ll }_{K,c,\hat{I}\pm }{H}^{2\hat{I}\pm }$. In this paper, we improve on this larger sieve bound by showing that $|A|{\ll }_{K,c,\hat{I}\pm }{H}^{\hat{I}}\pm (\log H{)}^r$. 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 ${\mathcal{O}}_K$.