  Examining multiple ergodic averages whose iterates are integer parts of real
valued polynomials for totally ergodic systems, we provide various
characterizations of total joint ergodicity, meaning that an average converges
to the "expected" limit along every arithmetic progression. In particular, we
obtain a complete characterization when the number of iterates is at most two,
and disprove a conjecture of the first author. We also improve a result of
Frantzikinakis on joint ergodicity of Hardy field functions of at most
polynomial growth for totally ergodic systems, which extends a conjecture of
Bergelson-Moreira-Richter. Our method is to first use the methodology of
Frantzikinakis, which allows one to reduce the systems to rotations on abelian
groups without using deep tools from ergodic theory, then develop formulas for
integrals of exponential functions over subtori, and finally, compute
exponential sums for integer parts of real polynomials.
