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 $\{l_{1}p,l_{2}p,...,l_{k}p\}$. We then derive several multiple recurrence results and 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 $$\{n\in \mathbb{N}:d^{\ast }(\Lambda \cap (\Lambda +p_1(n))\cap (\Lambda +p_2(n))\cap (\Lambda +p_3(n)))>(d^{\ast }(\Lambda ){)}^4-\epsilon \}$$ has bounded gaps for "most" choices of integer polynomials $p_1,p_2,p_3$.

47 pages, Final version to appear in the Trans. Amer. Math. Soc