Year: 2026
Type: preprint
Abstract: We consider a flat torus with a marked line segment $β$ and prove linear asymptotics for closed curves on the torus disjoint from the interior of the line segment. We also consider segments that begin and terminate at the end points of $β$ whose interiors are disjoint from $β$ and prove $L\log L$ asymptotics. In all cases, after quotienting by the growth rate, we compute the constant explicitly, which differs depending on whether or not $β$ lies in a periodic direction on the torus. In particular, when $β$ lies along a non-periodic trajectory, we show the constant is $2/(|β| ζ(2))$. Moreover, we prove that as a function of the holonomy vector of $β$, the constants are continuous at the points where $β$ lies in a non-periodic direction.
Source: arXiv (Cornell University)
Authors: Aulicino, David, Masur, Howard
Language: English
Subfield: Mathematical Physics
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W7222996372