Quadratic asymptotics for cylinder and saddle-connection counts on all translation surfaces

Determine whether, for every translation surface, the number of families of saddle connections and the number of maximal cylinders of simple closed geodesics of length at most L each admit a quadratic asymptotic as L tends to infinity.

Background

The paper contrasts its quadratic counting theorem for zero-holonomy Z-covers of genus-2 square-tiled surfaces with the broader unresolved problem for arbitrary translation surfaces. Masur’s general bounds establish only that cylinder and saddle-connection counts are bounded above and below by constant multiples of πL², while the existence of an actual asymptotic with a well-defined leading constant is not known in full generality.

The problem concerns two related counting functions: families of saddle connections and maximal cylinders of simple closed geodesics whose lengths do not exceed L. Resolving it would extend known quadratic-asymptotic results beyond special classes such as Veech surfaces and almost-everywhere settings.