We study mean convergence of multiple ergodic averages, where the iterates arise from smooth functions of polynomial growth that belong to a Hardy field. Our results include all logarithmico-exponential functions of polynomial growth, such as the functions $t^{3/2}, tlog t$ and $e^{sqrt{log t}}$. We show that if all non-trivial linear combinations of the functions $a_1,...,a_k$ stay logarithmically away from rational polynomials, then the $L^2$-limit of the ergodic averages $frac{1}{N} sum_{n=1}^{N}f_1(T^{lfloor{a_1(n)}rfloor}x)cdots f_k(T^{lfloor{a_k(n)}rfloor}x)$ exists and is equal to the product of the integrals of the functions $f_1,...,f_k$ in ergodic systems, which establishes a conjecture of Frantzikinakis. Under some more general conditions on the functions $a_1,...,a_k$, we also find characteristic factors for convergence of the above averages and deduce a convergence result for weak-mixing systems.
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