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+ϵ 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.Comment: 36 page