An extension of the classical van der Waerden and Szemerédi theorems is proved for commuting operators whose exponents are polynomials. As a consequence, for example, one obtains the following result: Let S ⊆ Z l S\subseteq \mathbb {Z}^l be a set of positive upper Banach density, let p 1 ( n ) , … , p k ( n ) p_1(n),\dotsc ,p_k(n) be polynomials with rational coefficients taking integer values on the integers and satisfying p i ( 0 ) = 0 p_i(0)=0 , i = 1 , … , k ; i=1,\dotsc ,k; then for any v 1 , … , v k ∈ Z l v_1,\dotsc ,v_k\in \mathbb {Z}^l there exist an integer n n and a vector u ∈ Z l u\in \mathbb {Z}^l such that u + p i ( n ) v i ∈ S u+p_i(n)v_i\in S for each i ≤ k i\le k .

Link to this article: http://journals.cambridge.org/abstract_S0143385704000215How to cite this article: A. LEIBMAN (2005). Pointwise convergence of ergodic averages for polynomial sequences of translations on a nilmanifold.Abstract. We show that the orbit of a point on a compact nilmanifold X under the action of a polynomial sequence of translations on X is well distributed on the union of several sub-nilmanifolds of X. This implies that the ergodic averages of a continuous function on X along a polynomial sequence of translations on X converge pointwise.

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