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<p>We show that the statistics of the continued fraction expansion of a randomly chosen rational in the unit interval, with a fixed large denominator <i>q</i>, approaches the Gauss-Kuzmin statistics with polynomial rate in <i>q</i>. This improves on previous results giving the convergence without rate. As an application of this effective rate of convergence, we show that the statistics of a randomly chosen rational in the unit interval, with a fixed large denominator <i>q</i> and <i>prime</i> numerator, also approaches the Gauss-Kuzmin statistics. Our results are obtained as applications of improved non-escape of mass and equidistribution statements for the geodesic flow on the space <InlineEquation ID="IEq1"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="29_2025_1026_Article_IEq1.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="109" /> </InlineMediaObject> <EquationSource Format="TEX">\(\operatorname {SL}_2(\mathbb {R})/\operatorname {SL}_2(\mathbb {Z})\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mo>SL</mo> <mn>2</mn> </msub> <mrow> <mo stretchy="false">(</mo> <mi mathvariant="double-struck">R</mi> <mo stretchy="false">)</mo> </mrow> <mo stretchy="false">/</mo> <msub> <mo>SL</mo> <mn>2</mn> </msub> <mrow> <mo stretchy="false">(</mo> <mi mathvariant="double-struck">Z</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation>.</p>

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On the rate of convergence of continued fraction statistics of random rationals

  • 期刊论文

  • 出版模式：

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  • 发表日期：

    2025年3月12日

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  • Ofir David,
  • Taehyeong Kim,
  • Ron Mor,
  • Uri Shapira

摘要

We show that the statistics of the continued fraction expansion of a randomly chosen rational in the unit interval, with a fixed large denominator q, approaches the Gauss-Kuzmin statistics with polynomial rate in q. This improves on previous results giving the convergence without rate. As an application of this effective rate of convergence, we show that the statistics of a randomly chosen rational in the unit interval, with a fixed large denominator q and prime numerator, also approaches the Gauss-Kuzmin statistics. Our results are obtained as applications of improved non-escape of mass and equidistribution statements for the geodesic flow on the space \(\operatorname {SL}_2(\mathbb {R})/\operatorname {SL}_2(\mathbb {Z})\) SL 2 ( R ) / SL 2 ( Z ) .