Multiple ergodic averages for three polynomials and applications
Abstract
We find the smallest characteristic factor and a limit formula for the multiple ergodic averages associated to any family of three polynomials and polynomial families of the form $\{p,2p,...,kp\}$. We then derive several combinatorial implications, including an answer to a question of Brown, Graham, and Landman, and a generalization of the Polynomial Szemerédi Theorem of Bergelson and Leibman for families of three polynomials with not necessarily zero constant term. We also simplify and generalize a recent result of Bergelson, Host, and Kra, showing that for all $\epsilon>0$ and every subset of the integers $\Lambda$ the set $$ \big\{n\in\N\colon d^*\big(\Lambda\cap (\Lambda+p_1(n))\cap (\Lambda+p_2(n)) \cap (\Lambda+p_3(n))\big)>(d^*(\Lambda))^4\epsilon\big\} $$ has bounded gaps for "most" choices of integer polynomials $p_1,p_2,p_3$.
 Publication:

arXiv Mathematics eprints
 Pub Date:
 June 2006
 arXiv:
 arXiv:math/0606567
 Bibcode:
 2006math......6567F
 Keywords:

 Mathematics  Dynamical Systems;
 Mathematics  Combinatorics;
 37A45;
 37A30;
 28D05
 EPrint:
 45 pages