• DOI:10.4007/ANNALS.2012.175.2.2
• Corpus ID: 2172518

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}
}

• B. Green, T. Tao
• Published 22 September 2007
• Mathematics
• arXiv: Number Theory

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 periodic and "rational", and $(g'(a),g'(a+d),\ldots,g'(a + d(l-1)))$ is uniformly distributed (up to a specified error $\delta$) inside some subnilmanifold $G'/\Gamma'$ of $G/\Gamma$, for all sufficiently dense arithmetic progressions $a,a+d,\ldots,a+d(l-1)$ inside $\{1,..,N\}$.
Our bounds are uniform in $N$ and are polynomial in the error tolerance delta. In a subsequent paper we shall use this theorem to establish the Mobius and Nilsequences conjecture from our earlier paper "Linear equations in primes".