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We prove a quantitative estimate with a power saving error term for thenumber of points in a mapping class group orbit of Teichm\"uller space that liewithin a Teichm\"uller metric ball of given center and large radius. Estimatesof the same kind are also proved for sector and bisector counts. Theseestimates effectivize asymptotic counting results of Athreya, Bufetov, Eskin,and Mirzakhani.

Effective mapping class group dynamics I: Counting lattice points in Teichm\"uller space