The quantitative behaviour of polynomial orbits on nilmanifolds

TL;DR

This paper provides a quantitative analysis of polynomial orbits on nilmanifolds, showing their uniform distribution properties and a factorization that separates smooth, periodic, and equidistributed components, with bounds uniform in N.

Contribution

It introduces a quantitative version of Leibman's theorem, detailing the distribution of polynomial orbits on nilmanifolds with explicit bounds and a novel factorization approach.

Findings

Polynomial orbits are uniformly distributed in subnilmanifolds within specified error.

The factorization separates the orbit into smooth, periodic, and equidistributed parts.

Bounds are polynomial in the error tolerance and uniform in N.

Abstract

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".