The quantitative behaviour of polynomial orbits on nilmanifolds

Abstract

A theorem of Leibman asserts that a polynomial orbit $(g(1),g(2),g(3),\ldots)$ on a nilmanifold $G/Γ$ is always equidistributed in a union of closed sub-nilmanifolds of $G/Γ$. 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 $g = εg'γ$, where $ε(n)$ is "smooth", $γ(n)$ is periodic and "rational", and $(g'(a),g'(a+d),\ldots,g'(a + d(l-1)))$ is uniformly distributed (up to a specified error $δ$) inside some subnilmanifold $G'/Γ'$ of $G/Γ$, 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".

62pp. Appeared as Ann. Math. 175 (2012), no. 2, 465--540. August 2015: footnote added in Section 8 to explain an error in the multidimensional case and to link to an erratum resolving this issue