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The main results of this paper are to prove bounds for ergodic averages fornilflows on general higher step nilmanifolds. Under Diophantine condition onthe frequency of a toral projection of the flow, we prove that almost allorbits become equidistributed at the polynomial speed. We analyze the exponentwith the speed of decay which is determined by the number of steps andstructure of general nilpotent Lie algebras. The main result follows from thetechnique over controlling scaling operators in irreducible representations andmeasure estimation on close return orbit on general nilmanifolds.

Effective equidistribution for generalized higher step nilflows