MULTIFRACTAL ANALYSIS OF THE DIVERGENCE POINTS ASSOCIATED WITH THE GROWTH OF DIGITS IN ENGEL EXPANSIONS

Lei Shang () and Yao Chen
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Lei Shang: College of Sciences, Nanjing Agricultural University, 210095 Nanjing, P. R. China
Yao Chen: College of Sciences, Nanjing Agricultural University, 210095 Nanjing, P. R. China

FRACTALS (fractals), 2025, vol. 33, issue 03, 1-8

Abstract: 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).

Keywords: Multifractal Analysis; Divergence Points; Engel Expansions (search for similar items in EconPapers)
Date: 2025
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DOI: 10.1142/S0218348X24501330

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