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

TL;DR

This paper investigates the extreme value distribution of the unipotent flow on the modular surface, deriving explicit formulas and asymptotic behavior for the distribution of deepest cusp excursions.

Contribution

It establishes the existence of a continuous distribution function for cusp excursions and provides explicit formulas and asymptotics, advancing understanding of unipotent flow extremes.

Findings

Derived explicit formulas for the distribution function F(r)

Proved asymptotic behavior of F(r) as r approaches -infinity

Established the existence of a continuous distribution for cusp excursions

Abstract

We study an extreme value distribution for the unipotent flow on the modular surface $\mathrm{SL}_2(\mathbb{R})/\mathrm{SL}_2(\mathbb{Z})$. 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 $r \in [-\frac{1}{2} \log 2, \infty)$, and establish asymptotic behavior of $F(r)$ as $r \to -\infty$.