Equidistribution of dilations of polynomial curves in nilmanifolds

In this paper we study the asymptotic behaviour under dilations of probability measures supported on polynomial curves in nilmanifolds. We prove, under some mild conditions, the effective equidistribution of such measures to the Haar measure. We also formulate a mean ergodic theorem for $\mathbb{R}^n$-representations on Hilbert spaces, restricted to a moving phase of low dimension. Furthermore, we bound the necessary dilation of a given smooth curve in $\mathbb{R}^n$ so that the canonical projection onto $\mathbb{T}^n$ is $\varepsilon$-dense.