Measure bound for translation surfaces with short saddle connections

We prove that any ergodic $S{L}_2(R)$-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 ${ϵ}_1,{ϵ}_2$ is $O({ϵ}_1^2{ϵ}_2^2)$. 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.