Distribution of values of bounded generalized polynomials

Abstract

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: Z^d → R^l has a representation u(n) = f(ϕ(n)x), n ∈ Z^d, where f is a piecewise polynomial function on a compact nilmanifold X, x ∈ X, and ϕ is an ergodic Z^d-action by translations on X. This fact is used to show that the sequence u(n), n ∈ Z^d, is well distributed on a piecewise polynomial surface $\mathcal{S}\subset\mathbf{R}^{l}$ (with respect to the Borel measure on $\mathcal{S}$ that is the image of the Lebesgue measure under the piecewise polynomial function defining $\mathcal{S}$). 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.