Following an approach presented by N. Frantzikinakis, B. Host and B. Kra, we show that the parameters in the multidimensional Szemer\'edi theorem for closest integer polynomials have non-empty intersection with the set of shifted primes $\mathbb{ P}-1$ (or similarly of $\mathbb{ P}+1$). Using the Furstenberg Correspondence Principle, we show this result by recasting it as a polynomial multiple recurrence result in measure ergodic theory. Furthermore, we obtain integer part polynomial convergence results by the same method, which is a transference principle that enables one to deduce results for $\mathbb{ Z}$-actions from results for flows. We also give some applications of our approach on Gowers uniform sets.
Categories: math.DS math.CO math.NT