Borel-Bernstein theorem and Hausdorff dimension of sets in power-2-decaying Gauss-like expansion

Each x (0, 1] can be uniquely expanded as a power-2-decaying Gauss-like expansion, in the form of x=₈=₁^2^- (d₁ (x) +d₂ (x) ++dᵢ (x) ), dᵢ (x) N. Let: N R^+ be an arbitrary positive function. We are interested in the size of the set F () =\x (0, 1]: dₙ (x) (n) ~~i. m. ~n\. We prove a Borel-Bernstein theorem on the zero-one law of the Lebesgue measure of F (). We also obtain the Hausdorff dimension of F ().