Measure bound for translation surfaces with short saddle connections

Abstract

We prove that any ergodic $SL_2(R)$-invariant probability measure on a stratum of translation surfaces satisfies strong regularity: the measure of the set of surfaces with two non-parallel saddle connections of length at most $\epsilon_1, \epsilon_2$ is $O(\epsilon_1^2 \epsilon_2^2)$. We prove a more general theorem which works for any number of short saddle connections. The proof uses the multi-scale compactification of strata recently introduced by Bainbridge-Chen-Gendron-Grushevsky-M\"oller and the algebraicity result of Filip.