Polynomial Ergodic Averages Converge Rapidly: Variations on a Theorem of Bourgain

Let $L^2(X,Σ,μ,τ)$ be a measure-preserving system, with $τ$ a $\mathbb{Z}$-action. In this note, we prove that the ergodic averages along integer-valued polynomials, $P(n)$, $M_N(f):= \frac{1}{N}\sum_{n \leq N} τ^{P(n)} f$ converge pointwise for $f \in L^2(X)$. We do so by proving that, for $r>2$, the $r$-variation, $\mathcal{V}^r(M_N(f))$, extends to a bounded operator on $L^2$. We also prove that our result is sharp, in that $\mathcal{V}^2(M_N(f))$ is an unbounded operator on $L^2$.

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