Bourgain's $L^2$ pointwise ergodic theorem over function fields

Abstract

We prove a function-field analogue of Bourgain's $L^2$ pointwise ergodic theorem. Let $q$ be a power of a prime $p$, let ${\mathbb{F}}_q[t]$ be the ring of polynomials over the finite field ${\mathbb{F}}_q$, and let ${\mathbb{F}}_q[t][u]$ be the ring of polynomials over ${\mathbb{F}}_q[t]$. Let $T^{(1)},\dots ,T^{(\ell )}$ be commuting, measure-preserving ${\mathbb{F}}_q[t]$-actions on a $\ddot{I}ƒ$-finite measure space $(X,\hat{I}¼)$, and let $P_1,\dots ,P_{\ell }\in {\mathbb{F}}_q[t][u]\setminus \{0\}$. Define a sequence of operators $(A_n{)}_{n\in \mathbb{N}}$ by \[ A_n g(x):=\frac{1}{q^n}\sum_{\substack{f\in \mathbb{F}_q[t]\\°f<n}} g\left(T^{(1)}_{P_1(f)}\cdots T^{(\ell)}_{P_\ell(f)}x\right) \qquad \left( g\in L^2(X),\,\,x\in X\right). \] We prove that $(A_n{)}_{n\in \mathbb{N}}$ satisfies an $L^2$ oscillation ergodic theorem: \[ \sup_{\substack{n_1<\cdots <n_{t_0}\\ t_0\in \mathbb{N}}} \left( \int_X \sum_{j=1}^{t_0-1} \sup_{n_j\leq n<n_{j+1}} |A_ng(x)-A_{n_{j+1}}g(x)|^2 \,dμ(x) \right)^{1/2} \leq C_1\|g\|_{L^2(X)}\qquad \left( g\in L^2(X)\right), \] where the constant $C_1>0$ depends only on $P_1,\dots ,P_{\ell }$ and $q$. This in particular implies that the sequence $(A_{n}g(x){)}_{n\in \mathbb{N}}$ converges for almost every $x\in X$ and that $(A_n{)}_{n\in \mathbb{N}}$ satisfies an $L^2$ maximal inequality: \[ \big\|\sup_{n\in\mathbb{N}}|A_ng|\big\|_{L^2(X)} \leq C_2\|g\|_{L^2(X)} \qquad \left( g\in L^2(X)\right), \] where the constant $C_2>0$ depends only on $P_1,\dots ,P_{\ell }$ and $q$. Our tools include the circle method in function fields and refinements of Weyl sum estimates in this setting, further developing the work of Lê-Liu-Wooley and Champagne-Ge-Lê-Liu-Wooley. These refinements are of independent interest.

31 pages