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<p>Let <InlineEquation ID="IEq1"> <EquationSource Format="TEX">\(\cal{F}\)</EquationSource> <EquationSource Format="MATHML"><math display="block"> <mrow> <mi mathvariant="script">F</mi> </mrow> </math></EquationSource> </InlineEquation> be a countable collection of functions <i>f</i> defined on integers, with integer values, such that for every <InlineEquation ID="IEq2"> <EquationSource Format="TEX">\(f\in\cal{F}\)</EquationSource> <EquationSource Format="MATHML"><math display="block"> <mi>f</mi> <mo>∈</mo> <mrow> <mi mathvariant="script">F</mi> </mrow> </math></EquationSource> </InlineEquation>, <i>f</i>(<i>n</i>) → ∞ as <i>n</i> → ∞. This paper primarily investigates the Hausdorff dimension of the set of points whose digit sequences of the Engel expansion are strictly increasing and contain every finite pattern of <InlineEquation ID="IEq3"> <EquationSource Format="TEX">\(\cal{F}\)</EquationSource> <EquationSource Format="MATHML"><math display="block"> <mrow> <mi mathvariant="script">F</mi> </mrow> </math></EquationSource> </InlineEquation>, with applications demonstrated through representative examples.</p>

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Finite pattern problems related to Engel expansion

  • 期刊论文

  • 发表日期：

    2026年2月25日

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  • Chun-Yun Cao,
  • Yang Xiao

摘要

Let \(\cal{F}\) F be a countable collection of functions f defined on integers, with integer values, such that for every \(f\in\cal{F}\) f ∈ F , f(n) → ∞ as n → ∞. This paper primarily investigates the Hausdorff dimension of the set of points whose digit sequences of the Engel expansion are strictly increasing and contain every finite pattern of \(\cal{F}\) F , with applications demonstrated through representative examples.