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Published online by Cambridge University Press: 07 October 2019
Let 
$X$ be a finite-dimensional connected compact abelian group equipped with the normalized Haar measure 
$\unicode[STIX]{x1D707}$. We obtain the following mean ergodic theorem over ‘thin’ phase sets. Fix 
$k\geq 1$ and, for every 
$n\geq 1$, let 
$A_{n}$ be a subset of 
$\mathbb{Z}^{k}\cap [-n,n]^{k}$. Assume that 
$(A_{n})_{n\geq 1}$ has 
$\unicode[STIX]{x1D714}(1/n)$ density in the sense that 
$\lim _{n\rightarrow \infty }(|A_{n}|/n^{k-1})=\infty$. Let 
$T_{1},\ldots ,T_{k}$ be ergodic automorphisms of 
$X$. We have for any 
$$\begin{eqnarray}\frac{1}{|A_{n}|}\mathop{\sum }_{(n_{1},\ldots ,n_{k})\in A_{n}}f_{1}(T_{1}^{n_{1}}(x))\cdots f_{k}(T_{k}^{n_{k}}(x))\stackrel{L_{\unicode[STIX]{x1D707}}^{2}}{\longrightarrow }\int f_{1}\,d\unicode[STIX]{x1D707}\cdots \int f_{k}\,d\unicode[STIX]{x1D707},\end{eqnarray}$$
$f_{1},\ldots ,f_{k}\in L_{\unicode[STIX]{x1D707}}^{\infty }$. When the 
$T_{i}$ are ergodic epimorphisms, the same conclusion holds under the further assumption that 
$A_{n}$ is a subset of 
$[0,n]^{k}$ for every 
$n$. The density assumption on the 
$A_{i}$ is necessary. Immediate applications include certain Poincaré style recurrence results.