Restricted Diophantine approximation in quadratic number fields

Abstract: 

Dirichlet's approximation theorem tells us that, given any irrational $\alpha$, the inequality $|\alpha-a/q| \le q^{-2}$ is satisfied for infinitely many fractions $a/q$ of coprime integers $a$ and $q$. The more difficult problem of approximating irrationals by fractions $a/p$ with denominator $p$ restricted to primes has a long history starting with Vinogradov, who managed to handle linear exponential sums over primes unconditionally, and culminating in Matomaki's work making use of bounds for averages of Kloosterman sums which marks the limit of the currently available technology. We report about recent progress on the same problem in quadratic number fields of class number 1. This is joint work with Marc Technau (University of Graz) in the case of imaginary quadratic fields and Dwaipayan Mazumder (RKMVERI) in the case of real quadratic fields.