Equidistribution of saddle connections on translation surfaces

Convergence of Siegel–Veech constants

“…It is a recurrence-type result which controls the length of the shortest saddle connection, on average, over translation surfaces on a large "circle" centered at any X. Here we deduce the proposition directly from a more general result proved in [Doz17]; in that paper the more general result is used to study the distribution of angles of saddle connections.…”

“…Proof of Proposition 1.1. This is a special case of Proposition 2.1 in [Doz17]. That Proposition involves integrating over any subinterval I ⊂ [0, 2π]; the above is simply the case when I equals [0, 2π].…”

“…If we keep track of the c max coming from the proof of Theorem 1.2, we find it grows at most exponentially in the genus, but it seems hard to do better than exponential using our method. The exponential nature of the method arises from an induction on the complexity of certain "complexes" of saddle connections in the proof of the generalization of Proposition 1.1 given in [Doz17].…”

Uniform distribution of saddle connection lengths

“…for m almost every ω. Dozier [Doz17b] proved analogues of these results for saddle connections with holonomies in certain sectors of directions.…”