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

Abstract

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 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 $F_d(S)={z:\ a_n(z)\in S\ \text{for all } n,\ |a_n(z)|\to\infty}$ satisfies $\dim_H F_d(S)=\tau(S)/2$. More generally, for any cutoff function $f(n)\to\infty$, the set $F_d(S,f)={z\in F_d(S):\ |a_n(z)|\le f(n)\ \text{for all } n}$ is either empty or has the same Hausdorff dimension $\tau(S)/2$. 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 $L'evy$- and Khinchine-type laws.