  Let $(X,\mathcal{B},\mu,T)$ be a probability-preserving system with $X$
compact and $T$ a homeomorphism. We show that if every point in $X\times X$ is
two-sided recurrent, then $h_{\mu}(T)=0$, resolving a problem of Benjamin
Weiss, and that if $h_{\mu}(T)=\infty$ then every full-measure set in $X$
contains mean-asymptotic pairs (i.e. the associated process is not tight),
resolving a problem of Ornstein and Weiss.
