For nearest-integer complex continued fractions over the five Euclidean imaginary quadratic fields, the set with |a_n| >= u_n infinitely often has full or zero Lebesgue measure according as sum u_n^{-2} diverges or converges, and digit-restricted sets F_d(S) have Hausdorff dimension tau(S)/2.
Zur metrischen theorie der diophantischen approximationen.Prace Matematyczno-Fizyczne, 36(1):91–106, 1928
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Borel--Bernstein and Hirst-type Theorems for Nearest-Integer Complex Continued Fractions over Euclidean Imaginary Quadratic Fields
For nearest-integer complex continued fractions over the five Euclidean imaginary quadratic fields, the set with |a_n| >= u_n infinitely often has full or zero Lebesgue measure according as sum u_n^{-2} diverges or converges, and digit-restricted sets F_d(S) have Hausdorff dimension tau(S)/2.