  Let $S$ and $T$ be measure-preserving transformations of a probability space
$(X,\mathcal{B},\mu)$. Let $f$ be a bounded measurable functions, and consider
the integrals of the corresponding `double' ergodic averages:
\[\frac{1}{n}\sum_{i=0}^{n-1} \int f(S^ix)f(T^ix)\ d\mu(x) \qquad (n\ge 1).\]
We provide a new construction of diverse examples for which these integrals do
not converge as $n\to\infty$. These include examples in which $S$ and $T$ are
rigid, and hence have entropy zero, answering a question of Frantzikinakis and
Host. We begin with a corresponding construction of pairs of unitary operators
on a complex Hilbert space, and then construct transformations of a Gaussian
measure space from them.
