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.

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      JOURNAL = {Pure Appl. Math. Q.},
      FJOURNAL = {Pure and Applied Mathematics Quarterly},
      VOLUME = {2},
      YEAR = {2006},
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      PAGES = {395--433},
      ISSN = {1558-8599},
      MRCLASS = {11N13 (11B25 37A45)},
      MRNUMBER = {2251475},
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      NOTE={Proceedings of the NATO Advanced Study Institute on Equidistribution in Number Theory, Montreal, Canada, 11--22 July 2005},
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      TITLE = {Universal characteristic factors and {F}urstenberg averages},
      JOURNAL = {J. Amer. Math. Soc.},
      FJOURNAL = {Journal of the American Mathematical Society},
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      YEAR = {2007},
      NUMBER = {1},
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      MRCLASS = {37A30 (28D05 37A25)},
      MRNUMBER = {2257397},
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      DOI = {10.1090/S0894-0347-06-00532-7},
      ZBLNUMBER = {1198.37014},
      }

Authors

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