Uniform distribution in nilmanifolds along functions from a Hardy field

We study equidistribution properties of translations on nilmanifolds along functions of polynomial growth from a Hardy field. More precisely, if $X=G/Gamma$ is a nilmanifold, $a_1,ldots,a_kin G$ are commuting nilrotations, and $f_1,ldots,f_k$ are functions of polynomial growth from a Hardy field then we show that $bullet$ the distribution of the sequence $a_1^{f_1(n)}cdotldotscdot a_k^{f_k(n)}Gamma$ is governed by its projection onto the maximal factor torus, which extends Leibmans Equidistribution Criterion form polynomials to a much wider range of functions; and $bullet$ the orbit closure of $a_1^{f_1(n)}cdotldotscdot a_k^{f_k(n)}Gamma$ is always a finite union of sub-nilmanifolds, which extends some of the previous work of Leibman and Frantzikinakis on this topic.