DIOPHANTINE APPROXIMATION WITH GAUSSIAN PRIMES

Abstract

Abstract In this paper we prove that the exact analogue of the author’s work with real irrationals and rational primes (G. Harman, On the distribution of $\alpha p$ modulo one II, Proc. London Math. Soc. (3) 72, 1996, 241–260) holds for approximating $\alpha \in \mathbb{C}\setminus \mathbb{Q}[i]$ with Gaussian primes. To be precise, we show that for such $\alpha $ and arbitrary complex $\beta $ there are infinitely many solutions in Gaussian primes $p$ to $$\begin{equation*} ||\alpha p + \beta|| <| p|^{-7/22}, \end{equation*}$$where $||\cdot ||$ denotes distance to a nearest member of $\mathbb{Z}[i]$. We shall, in fact, prove a slightly more general result with the Gaussian primes in sectors, and along the way improve a recent result due to Baier (S. Baier, Diophantine approximation on lines in $\mathbb{C}^2$ with Gaussian prime constraints, Eur. J. Math. 3, 2017, 614–649).