A theorem of Leibman asserts that a polynomial orbit (g(n)Γ)n∈ℤ on a nilmanifold G/Γ is always equidistributed in a union of closed sub-nilmanifolds of G/Γ. In this paper we give a quantitative version of Leibman’s result, describing the uniform distribution properties of a finite polynomial orbit (g(n)Γ)n∈[N] in a nilmanifold. More specifically we show that there is a factorisation g=εg′γ, where ε(n) is “smooth,” (γ(n)Γ)n∈ℤ is periodic and “rational,” and (g′(n)Γ)n∈P is uniformly distributed (up to a specified error δ) inside some subnilmanifold G′/Γ′ of G/Γ for all sufficiently dense arithmetic progressions P⊆[N].
Our bounds are uniform in N and are polynomial in the error tolerance δ. In a companion paper we shall use this theorem to establish the Möbius and Nilsequences conjecture from an earlier paper of ours.