On the exact rate of convergence of digits in Engel expansions

Abstract

Let ϕ:N→R+ be a function satisfying ϕ(n)→∞ as n→∞. Denote by 〈d1(x),d2(x),⋯,dn(x),⋯〉 the Engel expansion of an irrational number x∈(0,1). We study the Baire classification and Hausdorff dimension ofDϕ(α,β):={x∈(0,1)﹨Q:liminfn→∞log⁡dn(x)−nϕ(n)=α,limsupn→∞log⁡dn(x)−nϕ(n)=β} for α,β∈[−∞,∞] with α≤β. Under the condition that ϕ(n+1)−ϕ(n)→0 as n→∞, we show that the set Dϕ(α,β) is residual if and only if α=−∞ and β=∞. Under the conditions that ϕ is increasing and ϕ(n+1)−ϕ(n)→0 as n→∞, we prove that Dϕ(α,β) has full Hausdorff dimension for any −∞≤α≤β≤∞, which improves the result of Liu and Wu (2003). We also do some similar analyses for the gap of consecutive digits.