Multi-Frequency Oscillation Estimates Arising in Pointwise Ergodic Theory
Ben Krause
TL;DR
This work establishes essentially optimal $L^p$-bounds for variational variants of multi-frequency maximal Fourier multipliers arising in pointwise ergodic theory. By combining a smooth-cutoff reduction, metric-entropy chaining, and vector-valued oscillation inequalities, the authors obtain strong $L^2$ and $L^p$ control of the $r$-variation for the multi-frequency projections, and an endpoint weak-type result via a Nazarov–Oberlin–Thiele Calderón–Zygmund decomposition. A central corollary shows almost everywhere convergence of ergodic averages of the form $ rac{1}{N}\sum_{n\, le N} T^{ loor{P(n)}} f$ for polynomials with at least one irrational coefficient, in the expected $p$-range, with an appendix sketching a broader extension to all $p$ using higher-dimensional transference. The work blends harmonic-analytic and ergodic-analytic techniques, including Gauss-sum analyses and major/minor arc decomposition, to advance understanding of polynomial-trajectory ergodic averages and their oscillatory behavior.
Abstract
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 \[ \frac{1}{N} \sum_{n \leq N} T^{\lfloor P(n) \rfloor} f, \; \; \; f \in L^p(X,μ), \ P \in \mathbb{R}[\cdot], \] for any $σ$-finite measure space equipped with a measure-preserving transformation, $T:X \to X$, whenever $1 < p \leq \infty$ if $P$ is linear, and $4/3 < p \leq \infty$ otherwise.