An $Ω$-Result for the Counting of Geodesic Segments in the Hyperbolic Plane
Marios Voskou
TL;DR
This work addresses the Omega-type lower bound for the mollified error in counting hyperbolic double cosets near a fixed geodesic for a cocompact Fuchsian group. The author employs mollification and Lekkas' relative trace formula to express the mollified error as a spectral sum, then analyzes the resulting coefficients via hypergeometric expansions and a resonance argument to force constructive interference among spectral terms. By constructing resonant values of the hyperbolic parameter R, the argument shows that the mollified error tilde E(X,l) satisfies tilde E(X,l) = Ω_δ( X^{1/2} (log log X)^{1/4−δ} ) for any δ > 0, with X = cosh R. This result mirrors analogous Omega bounds in the classical lattice counting problem and demonstrates the limits of current upper bounds in the hyperbolic double coset setting, highlighting a resonance mechanism in spectral sums.
Abstract
Let $Γ$ be a cocompact Fuchsian group, and $l$ a fixed closed geodesic. We study the counting of those images of $l$ that have a distance from $l$ less than or equal to $R$. We prove an $Ω$-result for the error term in the asymptotic expansion of the counting function. More specifically, we prove that the error term is equal to $Ω_δ\left(X^{1/2}\left(\log{\log{X}}\right)^{1/4-δ} \right)$, where $X=\cosh{R}$.