Published online by Cambridge University Press: 06 November 2014
We study double averages along orbits for measure-preserving actions of $\mathbb{A}^{{\it\omega}}$, the direct sum of countably many copies of a finite abelian group
$\mathbb{A}$. We show an
$\text{L}^{p}$ norm-variation estimate for these averages, which in particular re-proves their convergence in
$\text{L}^{p}$ for any finite
$p$ and for any choice of two
$\text{L}^{\infty }$ functions. The result is motivated by recent questions on quantifying convergence of multiple ergodic averages.