Integer part independent polynomial averages and applications along primes

Exploiting the equidistribution properties of polynomial sequences, following the methods developed by Leibman (2005) and Frantzikinakis (2009, 2010), we show that the ergodic averages with iterates given by the integer parts of strongly independent real valued polynomials converge in the mean to the expected limit. These results have, via Furstenberg's correspondence principle, immediate combinatorial applications, while combining these results with methods of Frantzikinakis et al. (2013) and Koutsogiannis (2018) we get the expected limits and combinatorial results for multiple averages for a single sequence, as well as for several sequences along prime numbers.