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<p>We study an extreme value distribution for the unipotent flow on the modular surface <InlineEquation ID="IEq3"> <EquationSource Format="TEX">\(\textrm{SL}_2({\mathbb {R}})/\textrm{SL}_2({\mathbb {Z}})\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mtext>SL</mtext> <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> <mtext>SL</mtext> <mn>2</mn> </msub> <mrow> <mo stretchy="false">(</mo> <mi mathvariant="double-struck">Z</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation>. Using tools from homogenous dynamics and geometry of numbers we prove the existence of a continuous distribution function <i>F</i>(<i>r</i>) for the normalized deepest cusp excursions of the unipotent flow. We find closed analytic formulas for <i>F</i>(<i>r</i>) for <InlineEquation ID="IEq4"> <EquationSource Format="TEX">\(r \in [-\frac{1}{2} \log 2, \infty )\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>r</mi> <mo>∈</mo> <mo stretchy="false">[</mo> <mo>-</mo> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> <mo>log</mo> <mn>2</mn> <mo>,</mo> <mi>∞</mi> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation>, and establish asymptotic behavior of <i>F</i>(<i>r</i>) as <InlineEquation ID="IEq5"> <EquationSource Format="TEX">\(r \rightarrow -\infty \)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>r</mi> <mo stretchy="false">→</mo> <mo>-</mo> <mi>∞</mi> </mrow> </math></EquationSource> </InlineEquation>.</p>

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On an extreme value law for the unipotent flow on \(\textrm{SL}_2({\mathbb {R}})/\textrm{SL}_2({\mathbb {Z}})\)

  • 期刊论文

  • 发表日期：

    2026年2月28日

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  • Maxim Kirsebom,
  • Keivan Mallahi-Karai

摘要

We study an extreme value distribution for the unipotent flow on the modular surface \(\textrm{SL}_2({\mathbb {R}})/\textrm{SL}_2({\mathbb {Z}})\) SL 2 ( R ) / SL 2 ( Z ) . Using tools from homogenous dynamics and geometry of numbers we prove the existence of a continuous distribution function F(r) for the normalized deepest cusp excursions of the unipotent flow. We find closed analytic formulas for F(r) for \(r \in [-\frac{1}{2} \log 2, \infty )\) r ∈ [ - 1 2 log 2 , ∞ ) , and establish asymptotic behavior of F(r) as \(r \rightarrow -\infty \) r → - ∞ .