Multiple ergodic averages for three polynomials and applications

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 $p_1,p_2,p_3$.