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We investigate the distribution of $\alpha p$ modulo one in imaginaryquadratic number fields $\mathbb{K}\subset\mathbb{C}$ with class number one,where $p$ is restricted to prime elements in the ring of integers $\mathcal{O}= \mathbb{Z}[\omega]$ of $\mathbb{K}$. In analogy to classical work due to R.C. Vaughan, we obtain that the inequality $\lVert\alpha p\rVert_\omega <\mathrm{N}(p)^{-1/8+\epsilon}$ is satisfied for infinitely many $p$, where$\lVert\varrho\rVert_\omega$ measures the distance of $\varrho\in\mathbb{C}$ to$\mathscr{O}$ and $\mathrm{N}(p)$ denotes the norm of $p$.  The proof is based on Harman's sieve method and employs number fieldanalogues of classical ideas due to Vinogradov. Moreover, we introduce asmoothing which allows us to make conveniently use of the Poisson summationformula.

On the distribution of $\alpha p$ modulo one in imaginary quadratic number fields with class number one