Borel--Bernstein and Hirst-type Theorems for Nearest-Integer Complex Continued Fractions over Euclidean Imaginary Quadratic Fields

For each $d \in {1,2,3,7,11}$, let $T_d$ be the nearest-integer complex continued fraction map associated with the Euclidean ring $\mathcal{O}*d$, and let $(a_n)$ be its digit sequence. We prove two metric results for this five-system family. First, for every sequence $(u_n)*{n\ge 1}$ with $u_n \ge 1$, the set of points for which $|a_n| \ge u_n$ for infinitely many $n$ has full or zero normalized Lebesgue measure according as $\sum_{n=1}^\infty u_n^{-2}$ diverges or converges. This gives a unifi