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/ Γ is a nilmanifold, a 1 , …, a k ∈ G are commuting nilrotations, and f 1 , …, f k are functions of polynomial growth from a Hardy field then we show that the distribution of the sequence a 1 f 1 ( n ) ⋯ a k f k ( n ) Γ is governed by its projection onto the maximal factor torus, which extends Leibman’s Equidistribution Criterion from polynomials to a much wider range of functions; and the orbit closure of a 1 f 1 ( n ) ⋯ a k f k ( n ) Γ is always a finite union of sub-nilmanifolds, which extends previous work of Leibman and Frantzikinakis on this topic.