The quantitative behaviour of polynomial orbits on nilmanifolds

Abstract

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 subnilmanifolds 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.