Diophantine approximation with prime denominator in quadratic number fields under GRH

Abstract

Matomäki proved that if $\hat{I}\pm \in \mathbb{R}$ is irrational, then there are infinitely many primes $p$ such that $|\hat{I}\pm -a/p|\le p^{-4/3+\epsilon }$ for a suitable integer a. In this paper, we extend this result to all quadratic number fields under the condition that the Grand Riemann Hypothesis holds for their Hecke $L$-functions.

31 pages