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$E\subseteq \mathbb{Z}$
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
is an
$$\begin{eqnarray}\displaystyle \biggl\{(n_{1},\ldots ,n_{k})\in \mathbb{Z}^{k}:d^{\ast }\biggl(\mathop{\bigcap }_{e_{1},\ldots ,e_{k}\in \{0,1\}}(E-(e_{1}n_{1}+\cdots +e_{k}n_{k}))\biggr)>d^{\ast }(E)^{2^{k}}-\unicode[STIX]{x1D700}\biggr\} & & \displaystyle \nonumber\end{eqnarray}$$
$\text{AVIP}_{0}^{\ast }$
-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.