Equidistribution of dilated curves on nilmanifolds

TL;DR

This paper investigates the conditions under which dilated measures on nilmanifolds converge to the Haar measure, generalizing classical results and distinguishing between different types of equidistribution.

Contribution

It provides necessary and sufficient conditions for equidistribution of dilated measures on nilmanifolds, extending classical results to a broader setting.

Findings

Necessary and sufficient conditions for equidistribution established

Equidistribution along dilates characterized and distinguished

Conditions for equidistribution outside a density zero set derived

Abstract

Generalizing classic results for a family of measures in the torus, for a family $(\mu_t)_{t\geq 0}$ of measures defined on a nilmanifold $X$, we study conditions under which the family equidistributes, meaning conditions under which the measures $\mu_t$ converge as $t\to\infty$ in the weak$^\ast$ topology to the Haar measure on $X$. We give general conditions on a family of measures defined by a dilation process, showing necessary and sufficient conditions for equidistribution as the family dilates, along with conditions such that this holds for all dilates outside some set of density zero. Furthermore, we show that these two types of equidistribution are different.