Equidistribution of Dilations of Polynomial Curves in Nilmanifolds

TL;DR

This paper investigates how measures supported on polynomial curves in nilmanifolds become uniformly distributed under dilation, providing effective bounds and ergodic results.

Contribution

It establishes effective equidistribution results for dilated polynomial curves in nilmanifolds and introduces a mean ergodic theorem for low-dimensional phases.

Findings

Effective equidistribution of measures supported on polynomial curves

Bounds on dilation needed for density in torus projections

A mean ergodic theorem for $ ^n$-representations

Abstract

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, effective equidistribution of such measures to the Haar measure. We also formulate a mean ergodic theorem for $\R^n$-representations on Hilbert spaces, restricted to a moving phase of low dimension. Furthermore, we bound the necessary dilation of a given smooth curve in $ \R^n$ so that the canonical projection onto $ \T^n $ is $ \eps$-dense.