Pointwise multiple averages for systems with two commuting transformations

Abstract

We show that for every ergodic measure-preserving system $(X,{\mathcal{X}},\unicode[STIX]{x1D707},S,T)$ with commuting transformations $S$ and $T$, the average $$\begin{eqnarray}\frac{1}{N^{3}}\mathop{\sum }_{i,j,k=0}^{N-1}f_{0}(S^{j}T^{k}x)f_{1}(S^{i+j}T^{k}x)f_{2}(S^{j}T^{i+k}x)\end{eqnarray}$$ converges for $\unicode[STIX]{x1D707}$-almost every $x\in X$ as $N\rightarrow \infty$ for all $f_{0},f_{1},f_{2}\in L^{\infty }(\unicode[STIX]{x1D707})$. We also show that if $(X,{\mathcal{X}},\unicode[STIX]{x1D707},S,T)$ is an ergodic measurable distal system, then the average $$\begin{eqnarray}\frac{1}{N}\mathop{\sum }_{i=0}^{N-1}f_{1}(S^{i}x)f_{2}(T^{i}x)\end{eqnarray}$$ converges for $\unicode[STIX]{x1D707}$-almost every $x\in X$ as $N\rightarrow \infty$ for all $f_{1},f_{2}\in L^{\infty }(\unicode[STIX]{x1D707})$.