The main result in the present paper is the following theorem: If \(l,n\in\mathbb{N}\), \(\overline{q}_1,\ldots,\overline{q}_m:\mathbb{Z}\rightarrow\mathbb{Z}^l\) are polynomials with \(\overline{q}_i(0)=\overline{0}\) for \(i=1,2,\ldots,m\) and \(E\subset \mathbb{Z}^l\) has positive upper Banach density, \(d^{\ast}(E)>0\), then the set of integers \(n\) such that \[ d^{\ast}(E\cap (E-\overline{q}_1(n))\cap\cdots\cap(E-\overline{q}_m(n)))>0 \] has nonempty intersection with \(\mathbb{P}-1\) and \(\mathbb{P}+1\), where \(\mathbb{P}\) denotes the set of primes.
Actually the authors prove the ergodic version of the above theorem, since the two versions are equivalent by Furstenberg’s correspondence principle. They also show mean convergence results for the corresponding multiple ergodic averages over the primes, conditional on the convergence of the corresponding averages over the full set of natural numbers. In some cases these results are new and in some they generalize previous work by Wooley, Ziegler, Bergelson, and Leibman.