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 $|A \mod{\mathfrak{p}}|\leq \alpha |\mathcal{O}_{K}/\mathfrak{p}|$ for every prime ideal $\mathfrak{p}$ with $N\mathfrak{p}>c$ for some $\alpha \in (0,1)$. It follows from a larger sieve due to Ellenberg, Elsholtz, Hall and Kowalski that $|A| \ll_{K,c,\alpha}H^{2\alpha}$. In this paper, we improve on this larger sieve bound by showing that $|A|\ll_{K,c,\alpha}H^{\alpha}(\log H)^r$. 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 $\mathcal{O}_{K}$.