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Following an approach presented by N. Frantzikinakis, B. Host and B. Kra, weshow that the parameters in the multidimensional Szemer\'edi theorem forclosest integer polynomials have non-empty intersection with the set of shiftedprimes $\mathbb{P}-1$ (or similarly of $\mathbb{P}+1$). Using the FurstenbergCorrespondence Principle, we show this result by recasting it as a polynomialmultiple recurrence result in measure ergodic theory. Furthermore, we obtaininteger part polynomial convergence results by the same method, which is atransference principle that enables one to deduce results for$\mathbb{Z}$-actions from results for flows. We also give some applications ofour approach on Gowers uniform sets.

Closest integer polynomial multiple recurrence along shifted primes