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=∑ᵢ₌₁∞2-(d₁(x)+d₂(x)+⋯+dᵢ(x)), dᵢ(x)∈ N. Let φ:N→ R⁺ be an arbitrary positive function. We are interested in the size of the set F(φ)=∈ (0,1]:dₙ(x)≥ φ(n)~~i.m.~n\. We prove a Borel-Bernstein theorem on the zero-one law of the Lebesgue measure of F(φ). We also obtain the Hausdorff dimension of F(φ).