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

Let $L^2(X,\mathrm{\Sigma },\mu ,\tau )$ be a measure-preserving system, with $\tau$ a $\mathbb{Z}$-action. In this note, we prove that the ergodic averages along integer-valued polynomials, $P(n)$, $$M_N(f):=\frac{1}{N}\sum _{n\le N}{\tau }^{P(n)}f$$ 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 $L^2$.