Sets of large values of correlation functions for polynomial cubic configurations

We prove that for any set$E\subseteq \mathbb{Z}$with upper Banach density$d^{\ast }(E)>0$, the set ‘of cubic configurations’ in$E$is large in the following sense: for any$k\in \mathbb{N}$and any$\unicode[STIX]{x1D700}>0$, the set$$\begin{eqnarray}\displaystyle \biggl\{(n_{1},\ldots ,n_{k})\in \mathbb{Z}^{k}:d^{\ast }\biggl(\mathop{\bigcap }_{e_{1},\ldots ,e_{k}\in \{0,1\}}(E-(e_{1}n_{1}+\cdots +e_{k}n_{k}))\biggr)>d^{\ast }(E)^{2^{k}}-\unicode[STIX]{x1D700}\biggr\} & & \displaystyle \nonumber\end{eqnarray}$$is an$\text{AVIP}_{0}^{\ast }$-set. We then generalize this result to the case of ‘polynomial cubic configurations’$e_{1}p_{1}(n)+\cdots +e_{k}p_{k}(n)$, where the polynomials$p_{i}:\mathbb{Z}^{d}\longrightarrow \mathbb{Z}$are assumed to be sufficiently algebraically independent.