Pointwise Convergence of Ergodic Averages Along Hardy Field Sequences

Let $(X,\mu )$ be an arbitrary measure space equipped with a family of pairwise commuting measure preserving transformations $T_1,\dots ,T_m$. We prove that the ergodic averages $$A_{N;X}^{P_1,\dots ,P_m}f=\frac{1}{N}\sum _{n=1}^N{T}_1^{\lfloor P_1(n)\rfloor }\cdots T_m^{\lfloor P_m(n)\rfloor }f$$ converge pointwise $\mu$-almost everywhere as $N\to \mathrm{\infty }$ for any $f\in L^p(X)$ with $p>1$, where $P_1,\dots ,P_m$ are Hardy field functions which are “non-polynomial” and have distinct growth rates. To establish pointwise convergence we will prove a long-variational inequality, which will in turn prove that a maximal inequality holds for our averages. Additionally, by restricting the class of Hardy field functions to those with the same growth rate as $t^c$ for $c>0$ non-integer, we also prove full variational estimates. We are therefore able to provide quantitative bounds on the rate of convergence of exponential sums of the form $$\frac{1}{N}\sum _{n=1}^{N}e({\xi }_1\lfloor n^{c_1}\rfloor +\cdots +\lfloor n^{c_m}\rfloor )$$ where $0<c_1<\cdots <c_m$ are non-integer.