On the distribution of $αp$ modulo one in imaginary quadratic number fields with class number one

We investigate the distribution of αp modulo one in imaginary quadratic number fields K ⊂ C with class number one, where p is restricted to prime elements in the ring of integers O = Z[ω] of K. In analogy to classical work due to R. C. Vaughan, we obtain that the inequality ‖αp‖ _ω < N(p) ^−1/8+ɛ is satisfied for infinitely many p, where ‖ϱ‖ _ω measures the distance of ϱ ∈ C to O and N(p) denotes the norm of p. The proof is based on Harman’s sieve method and employs number field analogues of classical ideas due to Vinogradov. Moreover, we introduce a smoothing which allows us to make conveniently use of the Poisson summation formula.