The quantitative behaviour of polynomial orbits on nilmanifolds

Abstract

A theorem of Leibman asserts that a polynomial orbit $(g(n)\Gamma)_{n \in \mathbb{Z}}$ on a nilmanifold $G/\Gamma$ is always equidistributed in a union of closed sub-nilmanifolds of $G/\Gamma$. In this paper we give a quantitative version of Leibman’s result, describing the uniform distribution properties of a finite polynomial orbit $(g(n)\Gamma)_{n \in [N]}$ in a nilmanifold. More specifically we show that there is a factorisation $g = \varepsilon g’ \gamma$, where $\varepsilon(n)$ is “smooth,” $(\gamma(n)\Gamma)_{n \in \mathbb{Z}}$ is periodic and “rational,” and $(g'(n)\Gamma)_{n \in P}$ is uniformly distributed (up to a specified error $\delta$) inside some subnilmanifold $G’/\Gamma’$ of $G/\Gamma$ for all sufficiently dense arithmetic progressions $P \subseteq [N]$.

Our bounds are uniform in $N$ and are polynomial in the error tolerance $\delta$. In a companion paper we shall use this theorem to establish the Möbius and Nilsequences conjecture from an earlier paper of ours.