The quantitative behaviour of polynomial orbits on nilmanifolds

A theorem of Leibman asserts that a polynomial orbit $(g(1),g(2),g(3),\dots )$ on a nilmanifold $G\mathrm{/}\mathrm{\Gamma }$ is always equidistributed in a union of closed sub-nilmanifolds of $G\mathrm{/}\mathrm{\Gamma }$. In this paper we give a quantitative version of Leibman’s result, describing the uniform distribution properties of a finite polynomial orbit $(g(1),\dots ,g(N))$ in a nilmanifold. More specifically we show that there is a factorization $g = \epsilon g’\gamma$,where$\epsilon(n)$is\mathrm{"}smooth\mathrm{"},$\gamma(n)$ is periodic and “rational”, and $(g^{\mathrm{\prime }}(a),g^{\mathrm{\prime }}(a+d),\dots ,g^{\mathrm{\prime }}(a+d(l-1)))$ is uniformly distributed (up to a specified error $\delta$) inside some subnilmanifold $G^{\mathrm{\prime }}\mathrm{/}{\mathrm{\Gamma }}^{\mathrm{\prime }}$ of $G\mathrm{/}\mathrm{\Gamma }$, for all sufficiently dense arithmetic progressions $a,a+d,\dots ,a+d(l-1)$ inside $\{1,\mathrm{.}\mathrm{.},N\}$. 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”.