$\mathrm{CMC\text{-}1}$ surfaces in hyperbolic and de Sitter spaces with Cantor ends

We prove that on every compact Riemann surface $M$ there is a Cantor set $C \subset M$ such that $M \setminus C$ admits a proper conformal constant mean curvature one ($\mathrm{CMC\text{-}1}$) immersion into hyperbolic $3$-space $\mathbb{H}^3$. Moreover, we obtain that every bordered Riemann surface admits an almost proper $\mathrm{CMC\text{-}1}$ face into de Sitter $3$-space $\mathbb{S}_1^3$, and we show that on every compact Riemann surface $M$ there is a Cantor set $C \subset M$ such that $M \setminus C$ admits an almost proper $\mathrm{CMC\text{-}1}$ face into $\mathbb{S}_1^3$. These results follow from different uniform approximation theorems for holomorphic null curves in $\mathbb{C}^2 \times \mathbb{C}^*$ that we also establish in this paper.