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.