Lagrange spectrum for Diophantine approximations of complex numbers with real part equal to one half

Motivated by a theorem of A. Schmidt on the part of the complex Lagrange spectrum below $2$, we study the restricted Lagrange spectrum $L_{\frac{1}{2}+i\mathbb{R}}$ arising from the approximation of complex numbers of the form $\frac{1}{2}+i\alpha$, $\alpha \in \mathbb{R}\setminus \mathbb{Q}$ by Gaussian rationals $p\mathrm{/}q$ with $p,q\in \mathbb{Z}[i]$, $q\mathrm{\ne }0$. We show that this spectrum admits a description in terms of a dynamical spectrum associated with a real horseshoe. As a consequence, we obtain several fractal properties of $L_{\frac{1}{2}+i\mathbb{R}}$, such as continuity of the dimension function $t\mapsto {\dim }_H(L_{\frac{1}{2}+i\mathbb{R}}\cap (-\mathrm{\infty },t))$. We also prove that the set of complex numbers $z$ satisfying \begin{equation*} \left\lvert z-\frac{p}{q}\right\rvert\geq\frac{1}{2|q|^2}, \quad$\text{for all }$ p,q\in$\mathbb{Z}$[i], q\neq 0, \end{equation*} is uncountable. In fact, we show that this inequality holds for every complex number of the form $z=\frac{1}{2}(1+i\theta )$ where $\theta \in \mathbb{R}\setminus \mathbb{Q}$ is a root of one of Schmidt’s $C$-minimal forms.