Bracket polynomials, uniform distribution and dynamics

A classical theorem due to H. Weyl states that if P is a polynomial over R such that at least one of its coefficients (other than the constant term) is irrational, then the sequence P(n), n=1,2,..., is uniformly distributed mod 1. We will discuss some extensions of this theorem which involve "bracket polynomials" (also called"generalized polynomials"), that is, functions which are obtained from the conventional polynomials by the use of the greatest integer function, addition and multiplication. We will explain the role of dynamical systems on nil-manifolds in obtaining these results, and discuss the intrinsic connection between the generalized polynomials and the polynomial extensions of Szemeredi's theorem on arithmetic progressions.