Equidistribution of Sparse Sequences on Nilmanifolds

We study equidistribution properties of nil-orbits when the time parameter is restricted to the range of some sparse sequence that is not necessarily polynomial. For example, we show that if X = G/Γ is a nilmanifold, b ∈ G is an ergodic nilrotation, and c ∈ R \ Z is positive, then the sequence (b cΓ)n∈N is equidistributed in X. This is also the case when n c is replaced with a(n), where a(t) is a function that belongs to some Hardy field, has at most polynomial growth, and stays logarithmically away from polynomials, and when it is replaced with a random sequence of integers with sub-exponential growth. Similar results have been established by Boshernitzan when X is the circle.