<p>Let <InlineEquation ID="IEq1"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="11139_2025_1028_Article_IEq1.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="164" /> </InlineMediaObject> <EquationSource Format="TEX">$[a_1(x),a_2(x),a_3(x),\ldots ]$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mo stretchy="false">[</mo> <msub> <mi>a</mi> <mn>1</mn> </msub> <mrow> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mrow> <mo>,</mo> <msub> <mi>a</mi> <mn>2</mn> </msub> <mrow> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mrow> <mo>,</mo> <msub> <mi>a</mi> <mn>3</mn> </msub> <mrow> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mrow> <mo>,</mo> <mo>…</mo> <mo stretchy="false">]</mo> </mrow> </math></EquationSource> </InlineEquation> denote the continued fraction expansion of an irrational number <InlineEquation ID="IEq2"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="11139_2025_1028_Article_IEq2.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="66" /> </InlineMediaObject> <EquationSource Format="TEX">$x\in [0,1)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>x</mi> <mo>∈</mo> <mo stretchy="false">[</mo> <mn>0</mn> <mo>,</mo> <mn>1</mn> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation>. For any function <InlineEquation ID="IEq3"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="11139_2025_1028_Article_IEq3.gif" Format="GIF" Height="18" Rendition="HTML" Resolution="72" Type="Linedraw" Width="86" /> </InlineMediaObject> <EquationSource Format="TEX">$\psi :\mathbb {N}\rightarrow \mathbb {R}^+$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>ψ</mi> <mo>:</mo> <mi mathvariant="double-struck">N</mi> <mo stretchy="false">→</mo> <msup> <mrow> <mi mathvariant="double-struck">R</mi> </mrow> <mo>+</mo> </msup> </mrow> </math></EquationSource> </InlineEquation> satisfying <InlineEquation ID="IEq4"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="11139_2025_1028_Article_IEq4.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="82" /> </InlineMediaObject> <EquationSource Format="TEX">$\psi (n)\rightarrow \infty $</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>ψ</mi> <mo stretchy="false">(</mo> <mi>n</mi> <mo stretchy="false">)</mo> <mo stretchy="false">→</mo> <mi>∞</mi> </mrow> </math></EquationSource> </InlineEquation> as <InlineEquation ID="IEq5"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="11139_2025_1028_Article_IEq5.gif" Format="GIF" Height="10" Rendition="HTML" Resolution="72" Type="Linedraw" Width="58" /> </InlineMediaObject> <EquationSource Format="TEX">$n\rightarrow \infty $</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>n</mi> <mo stretchy="false">→</mo> <mi>∞</mi> </mrow> </math></EquationSource> </InlineEquation>, define <Equation ID="Equ8"> <MediaObject> <ImageObject Color="BlackWhite" FileRef="11139_2025_1028_Article_Equ8.gif" Format="GIF" Height="43" Rendition="HTML" Resolution="72" Type="Linedraw" Width="362" /> </MediaObject> <EquationSource Format="TEX">$\begin{aligned} E_{\sup }(\psi ):=\left\{ x\in [0,1)\backslash \mathbb {Q}:\, \limsup \limits _{n\rightarrow \infty }\frac{\log a_n(x)}{\psi (n)}=1\right\} . \end{aligned}$</EquationSource> <EquationSource Format="MATHML"><math display="block"> <mrow> <mtable> <mtr> <mtd columnalign="right"> <mrow> <msub> <mi>E</mi> <mo movablelimits="true">sup</mo> </msub> <mrow> <mo stretchy="false">(</mo> <mi>ψ</mi> <mo stretchy="false">)</mo> </mrow> <mo>:</mo> <mo>=</mo> <mfenced close="}" open="{"> <mrow> <mi>x</mi> <mo>∈</mo> <mrow> <mo stretchy="false">[</mo> <mn>0</mn> <mo>,</mo> <mn>1</mn> <mo stretchy="false">)</mo> </mrow> <mo stretchy="true">\</mo> <mi mathvariant="double-struck">Q</mi> <mo>:</mo> <mspace width="0.166667em" /> </mrow> <munder> <mo movablelimits="false">lim sup</mo> <mrow> <mi>n</mi> <mo stretchy="false">→</mo> <mi>∞</mi> </mrow> </munder> <mfrac> <mrow> <mo>log</mo> <msub> <mi>a</mi> <mi>n</mi> </msub> <mrow> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mrow> </mrow> <mrow> <mi>ψ</mi> <mo stretchy="false">(</mo> <mi>n</mi> <mo stretchy="false">)</mo> </mrow> </mfrac> <mo>=</mo> <mn>1</mn> </mfenced> <mo>.</mo> </mrow> </mtd> </mtr> </mtable> </mrow> </math></EquationSource> </Equation>In (Ramanujan J 56:891–909, 2021), the authors determined the Hausdorff dimension of <InlineEquation ID="IEq6"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="11139_2025_1028_Article_IEq6.gif" Format="GIF" Height="20" Rendition="HTML" Resolution="72" Type="Linedraw" Width="56" /> </InlineMediaObject> <EquationSource Format="TEX">$E_{\sup }(\psi )$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi>E</mi> <mo movablelimits="true">sup</mo> </msub> <mrow> <mo stretchy="false">(</mo> <mi>ψ</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation> under the condition that the limit <InlineEquation ID="IEq7"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="11139_2025_1028_Article_IEq7.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="112" /> </InlineMediaObject> <EquationSource Format="TEX">$\lim _{n\rightarrow \infty }\psi (n)/n$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mo movablelimits="true">lim</mo> <mrow> <mi>n</mi> <mo stretchy="false">→</mo> <mi>∞</mi> </mrow> </msub> <mi>ψ</mi> <mrow> <mo stretchy="false">(</mo> <mi>n</mi> <mo stretchy="false">)</mo> </mrow> <mo stretchy="false">/</mo> <mi>n</mi> </mrow> </math></EquationSource> </InlineEquation> exists. In this paper, we remove this hypothesis and provide a complete description of the Hausdorff dimension of <InlineEquation ID="IEq8"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="11139_2025_1028_Article_IEq8.gif" Format="GIF" Height="20" Rendition="HTML" Resolution="72" Type="Linedraw" Width="56" /> </InlineMediaObject> <EquationSource Format="TEX">$E_{\sup }(\psi )$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi>E</mi> <mo movablelimits="true">sup</mo> </msub> <mrow> <mo stretchy="false">(</mo> <mi>ψ</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation>.</p>

A remark on the exceptional sets of the Borel–Bernstein theorem in continued fractions

期刊论文
发表日期：

2025年1月31日

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Xiao Chen,
Junjie Li,
Lei Shang,
Xin Zeng

摘要

Let $[a_1(x),a_2(x),a_3(x),\ldots ]$ $[a_{1} (x), a_{2} (x), a_{3} (x), \dots]$ denote the continued fraction expansion of an irrational number $x\in [0,1)$ $x \in [0, 1)$ . For any function $\psi :\mathbb {N}\rightarrow \mathbb {R}^+$ $ψ : N \to R^{+}$ satisfying $\psi (n)\rightarrow \infty $ $ψ (n) \to \infty$ as $n\rightarrow \infty $ $n \to \infty$ , define $\begin{aligned} E_{\sup }(\psi ):=\left\{ x\in [0,1)\backslash \mathbb {Q}:\, \limsup \limits _{n\rightarrow \infty }\frac{\log a_n(x)}{\psi (n)}=1\right\} . \end{aligned}$ $\begin{matrix} E_{sup} (ψ) : = \{x \in [0, 1) \ Q : \underset{n \to \infty}{lim sup} \frac{log a_{n} (x)}{ψ (n)} = 1\} . \end{matrix}$ In (Ramanujan J 56:891–909, 2021), the authors determined the Hausdorff dimension of $E_{\sup }(\psi )$ $E_{sup} (ψ)$ under the condition that the limit $\lim _{n\rightarrow \infty }\psi (n)/n$ ${lim}_{n \to \infty} ψ (n) / n$ exists. In this paper, we remove this hypothesis and provide a complete description of the Hausdorff dimension of $E_{\sup }(\psi )$ $E_{sup} (ψ)$ .