Distribution of values of bounded generalized polynomials

A generalized polynomial is a real-valued function which is obtained from conventional polynomials by the use of the operations of addition, multiplication, and taking the integer part; a generalized polynomial mapping is a vector-valued mapping whose coordinates are generalized polynomials. We show that any bounded generalized polynomial mapping u: Zd → Rl has a representation u (n) = f (ϕ (n) x), n ∈ Zd, where f is a piecewise polynomial function on a compact nilmanifold X, x ∈ X, and ϕ is an ergodic Zd-action by translations on X. This fact is used to show that the sequence u (n), n ∈ Zd, is well distributed on a piecewise polynomial surface S^l (with respect to the Borel measure on S that is the image of the Lebesgue measure under the piecewise polynomial function defining S). As corollaries we also obtain a von Neumann-type ergodic theorem along generalized polynomials and a result on Diophantine approximations extending the work of van der Corput and of Furstenberg–Weiss.