Multi-Frequency Oscillation Estimates Arising in Pointwise Ergodic Theory

We prove essentially optimal $L^p(\mathbb{R})$-estimates for variational variants of the maximal Fourier multiplier operators considered by Bourgain in his work on pointwise convergence of polynomial ergodic averages. As a corollary of our methods, we are able to quickly extend a result of Bourgain, namely the pointwise convergence of ergodic averages of integer parts of real-variables polynomials, to a broader class of functions, previously considered in a wide range of contexts by Boshernitzan-Jones-Wierdl. Namely, the following averages converge almost everywhere $$\frac{1}{N}\sum _{n\le N}{T}^{\lfloor P(n)\rfloor }f,f\in L^p(X,\mu ),P\in \mathbb{R}[\cdot ],$$ for any $\sigma$-finite measure space equipped with a measure-preserving transformation, $T:X\to X$, whenever $1<p\le \mathrm{\infty }$ if $P$ is linear, and $4\mathrm{/}3<p\le \mathrm{\infty }$ otherwise.