# The quantitative behaviour of polynomial orbits on nilmanifolds

@article{Green2007TheQB, title={The quantitative behaviour of polynomial orbits on nilmanifolds}, author={Ben Green and Terence Tao}, journal={arXiv: Number Theory}, year={2007} }

A theorem of Leibman asserts that a polynomial orbit $(g(1),g(2),g(3),\ldots)$ 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 $(g(1),\ldots,g(N))$ in a nilmanifold. More specifically we show that there is a factorization $g = \epsilon g'\gamma$, where $\epsilon(n)$ is "smooth", $\gamma(n)$ is…

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