Equidistribution of dilated curves on nilmanifolds

Generalizing classic results for a family of measures in the torus, for a family $({\mu }_t{)}_{t\ge 0}$ 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 $t\to \mathrm{\infty }$ 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.