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 $\{l_1p,l_2p,\ldots ,l_kp\}$. 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 $\varepsilon >0$ and every subset of the integers $\Lambda$ the set \[ \big \{n\in \mathbb {N}\colon d^*\big (\Lambda \cap (\Lambda +p_1(n))\cap (\Lambda +p_2(n))\cap (\Lambda + p_3(n))\big )>(d^*(\Lambda ))^4-\varepsilon \big \} \] has bounded gaps for “most” choices of integer polynomials $p_1,p_2,p_3$.