A theorem of Leibman asserts that a polynomial orbit (g(1),g(2),g(3),...) on a nilmanifold G/L is always equidistributed in a union of closed sub-nilmanifolds of G/L. In this paper we give a quantitative version of Leibman's result, describing the uniform distribution properties of a finite polynomial orbit (g(1),...,g(N)) in a nilmanifold. More specifically we show that there is a factorization g = eg'p, where e(n) is "smooth", p(n) is periodic and "rational", and (g'(a),g'(a+d),...,g'(a + d...Expand abstract
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62 pages, to appear in Annals of Math. Small changes made in the
light of comments from the referee.