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 $\mathcal{O}\ast d$, and let $(a_n)$ be its digit sequence. We prove two metric results for this five-system family. First, for every sequence $(u_n)\ast n\ge 1$ with $u_n\ge 1$, the set of points for which $\mathrm{\mid }{a}_n\mathrm{\mid }\ge u_n$ for infinitely many $n$ has full or zero normalized Lebesgue measure according as ${\sum }_{n=1}^{\mathrm{\infty }}{u}_n^{-2}$ diverges or converges. This gives a unified Borel–Bernstein theorem, extending the Hurwitz case $d=1$ to all five Euclidean imaginary quadratic fields. Second, for any infinite set $S\subset {\mathcal{O}}_d$, if $\tau (S)$ denotes its convergence exponent, then the digit-restricted set $F_d(S)=z:a_n(z)\in S\text{ for all }n,\mathrm{\mid }{a}_n(z)\mathrm{\mid }\to \mathrm{\infty }$ satisfies ${\dim }_H{F}_d(S)=\tau (S)\mathrm{/}2$. More generally, for any cutoff function $f(n)\to \mathrm{\infty }$, the set $F_d(S,f)=z\in F_d(S):\mathrm{\mid }{a}_n(z)\mathrm{\mid }\le f(n)\text{ for all }n$ is either empty or has the same Hausdorff dimension $\tau (S)\mathrm{/}2$. The proof combines quantitative ergodic properties of the nearest-integer systems with a large-digit conformal iterated function subsystem that is $2$-decaying. We also obtain applications to sparse patterns, shrinking targets, and almost-sure $L^{\mathrm{\prime }}evy$- and Khinchine-type laws.