Multiple ergodic averages for three polynomials and applications

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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 } \{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 ε > 0 \varepsilon >0 and every subset of the integers Λ \Lambda the set \[ { n ∈ N : d ∗ ( Λ ∩ ( Λ + p 1 ( n ) ) ∩ ( Λ + p 2 ( n ) ) ∩ ( Λ + p 3 ( n ) ) ) > ( d ∗ ( Λ ) ) 4 − ε } \big \{n\in \mathbb {N}\colon d^*\big (\Lambda \cap (\Lambda +p_1(n))\cap (\Lambda +p_2(n))\cap (\Lambda + p_3(n))\big )>(d^*(\Lambda ))^4-\varepsilon \big \} \] has bounded gaps for “most” choices of integer polynomials p 1 , p 2 , p 3 p_1,p_2,p_3 .

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