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).

中文翻译：

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高硫素素对金黄花碱的近似

在本文中，我们证明了作者的作品与真实非理性和理性素数的确切相似之处（G.哈曼，关于以模II表示的$ \ alpha p $的分布，Proc.London Math.Soc。（3）72，1996 （241-260），用高斯素数逼近\ mathbb {C} \ setminus \ mathbb {Q} [i] $中的$ \ alpha \。确切地说，我们证明了对于这样的$ \ alpha $和任意复数$ \ beta $，在高斯素数$ p $到$$ \ begin {equation *} || \ alpha p + \ beta ||中有无数个解。<| p | ^ {-7/22}，\ end {equation *} $$，其中$ || \ cdot || $表示到$ \ mathbb {Z} [i] $的最近成员的距离。实际上，我们将用扇区中的高斯素数证明稍微更普遍的结果，并一路改善由于拜耳（S. Baier，用\\ mathbb {C} ^ 2 $高斯素数约束，Eur。J. 数学。2017年3月，第614–649页）。