Effective counting on translation surfaces

Abstract We prove an effective version of a celebrated result of Eskin and Masur: for any SL 2 ( R ) -invariant locus L of translation surfaces, there exists κ > 0 , such that for almost every translation surface in L , the number of saddle connections with holonomy vector of length at most T, grows like c T 2 + O ( T 2 − κ ) . We also provide effective versions of counting in sectors and in ellipses.