The quantitative behaviour of polynomial orbits on nilmanifolds


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.


Ben Green

Department of Pure Mathematics and Mathematical Statistics, Centre for Mathematical Sciences, Wilberforce Road, Cambridge CB3 0WA, England

Terence Tao

Department of Mathematics, University of California, Los Angeles, Los Angeles, CA 90095-1596