Romanov's theorem in number fields
Résumé
Romanov proved that a positive proportion of the integers has a representation as a sum of a prime and a power of an arbitrary fixed integer. We prove the analogous result for number fields. Furthermore we give an explicit lower bound for the lower density of Gaussian integers that have a representation as a sum of a Gaussian prime and a power of $1+i$. Finally, similar to Erd\H{o}s, we construct an explicit arithmetic progression of Gaussian integers with odd norm which do not have a representation of this type.