Integer part independent polynomial averages and applications along primes

Exploiting the equidistribution properties of polynomial sequences, following the methods developed by Leibman ("Pointwise Convergence of ergodic averages for polynomial sequences of translations on a nilmanifold. Ergodic Theory Dynam. Systems, 25 (2005) no. 1, 201-213") and Frantzikinakis ("Multiple recurrence and convergence for Hardy field sequences of polynomial growth. Journal d'Analyse Mathematique, 112 (2010), 79-135" and "Equidistribution of sparse sequences on nilmanifolds. Journal d'Analyse Mathematique, 109 (2009), 353-395") we show that the ergodic averages with iterates given by the integer part of real-valued strongly independent polynomials, converge in the mean to the "right"-expected limit. These results have, via Furstenberg's correspondence principle, immediate combinatorial applications while combining these results with methods from "The polynomial multidimensional Szemer\'edi theorem along shifted primes. Israel J. Math., 194 (2013), no. 1, 331-348" and "Closest integer polynomial multiple recurrence along shifted primes. Ergodic Theory Dynam. Systems, 1-20. doi:10.1017/etds.2016.40" we get the respective "right" limits and combinatorial results for multiple averages for a single sequence as well as for several sequences along prime numbers.