Multifractal Analysis Of The Divergence Points Associated With The Growth Of Digits In Engel Expansions

In this paper, we are concerned with the multifractal analysis of the divergence points in Engel expansions. Let x ∈ (0, 1) be an irrational number with Engel expansion 〈d1(x),d2(x),d3(x),…〉. For any 0 ≤ α ≤ β ≤∞, let D(α,β) := x ∈ (0, 1)∖ℚ :liminfn→∞log dn(x) log n = α,limsupn→∞log dn(x) log n = β. We prove that the Hausdorff dimension of D(α,β) is (α − 1)/α when 1 ≤ α ≤∞, and it is zero when 0 ≤ α < 1. This indicates that the Hausdorff dimension of D(α,β) is independent of β. A very different phenomenon is shown for the gap of consecutive digits. For any irrational number x ∈ (0, 1) and n ∈ ℕ, let Δn(x) := dn(x) − dn−1(x) with d0(x) ≡ 0. We derive that, for any 0 ≤ α ≤ β ≤∞, the set Δ(α,β) := x ∈ (0, 1)∖ℚ :liminfn→∞log Δn(x) log n = α,limsupn→∞log Δn(x) log n = β has Hausdorff dimension β/(β + 1).