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

We investigate the distribution of $\alpha p$ modulo one in imaginary quadratic 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}$$. In analogy to classical work due to R. C. Vaughan, we obtain that the inequality $\lVert\alpha p\rVert_\omega < $\mathrm{N}$(p)^{-1/8+\epsilon}$issatisfiedforinfinitelymany$p$,where$\lVert\varrho\rVert_\omega$measuresthedistanceof$\varrho\in$\mathbb{C}$$to$\mathscr{O}$and$$\mathrm{N}$(p)$denotesthenormof$p$. 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.