Sets of large values of polynomial multi-correlation functions

Abstract

Let $p_1,...,p_L\in Z[x_1,...,x_d]$ be non-constant polynomials with zero constant term. The ergodic theoretical proofs of the polynomial and the IP-polynomial Szemeredi theorems as well as some of the ergodic-theoretical and combinatorial consequences of the Density Polynomial Hales-Jewett conjecture (DPHJ) naturally lead to the study of sets of large returns which are defined as $$ R_\epsilon^{p_1,...,p_L}(A):=\{n\in Z^d\,|\,\mu(A\cap T_1^{-p_1( n)}A\cap\cdots\cap T_L^{-p_L(n)}A)>\mu^{L+1}(A)-\epsilon\}, $$ where the $T_j$'s are commuting and invertible $\mu$-preserving transformations, $A$ is measurable, and $\epsilon>0$. We obtain new results dealing with the sets of the form $R_\epsilon^{p_1,...,p_L}(A)$. Among other things, we show that every set of the form $R_\epsilon^{p_1,...,p_L}(A)$ is syndetic if and only if $p_1,...,p_L$ are linearly independent, answering a question asked by Frantzikinakis-Kuca. Moreover, the linear independence of $p_1,...,p_L$ implies that every set of the form $R_\epsilon^{p_1,...,p_L}(A)$ has the A-IP$^*$ property (="almost" IP$^*$ property), which is stronger than syndeticity. The following is one of the new combinatorial results obtained in this paper. Suppose that $p_1,...,p_L$ are linearly independent. For any set $E\subseteq Z^D$ with upper Banach density $d^*(E)>0$, any non-zero $v_1,..., v_L\in Z^D$, and any $\epsilon>0$, the set $$ S_\epsilon^{p_1,...,p_L}(E):=\{ n\in Z^d\,|\,d^*(E\cap (E-p_1(n)v_1)\cap \cdots\cap (E-p_L(n)v_L))>(d^*(E))^{L+1}-\epsilon\} $$ is A-IP$^*$. Furthermore, we prove that when $D>L>1$, this result is sharp: the A-IP$^*$ property cannot be upgraded to IP$^*$. The techniques developed in this paper lead to some additional applications. For example, we show that an amplified form of the IP-polynomial Szemeredi theorem conjectured by Bergelson- McCutcheon follows from the DPHJ.