Hausdorff dimensions of some exceptional sets in Engel expansions

Given any real number x∈(0,1], denote its Engel expansion by ∑n=1∞1d1(x)⋯dn(x), where {dj(x),j≥1} is a sequence of positive integers satisfying d1(x)≥2 and dj+1(x)≥dj(x) (j≥1). Suppose ϕ:N→R+ is a function satisfying ϕ(n+1)−ϕ(n)→∞ as n→∞. In this paper, we consider the setE(ϕ)={x∈(0,1]:limn→∞⁡log⁡dn(x)ϕ(n)=1}, and we quantify the size of E(ϕ) in the sense of Hausdorff dimension. As applications, we get the Hausdorff dimensions of the sets {x∈(0,1]:limn→∞⁡log⁡dn(x)nβ=γ} and {x∈(0,1]:limn→∞⁡log⁡dn(x)τn=η}, where β>1,γ>0 and τ>1,η>0.