  Consider a countable amenable group acting by homeomorphisms on a compact
metrizable space. Chung and Li asked if expansiveness and positive entropy of
the action imply existence of an off-diagonal asymptotic pair. For algebraic
actions of polycyclic-by-finite groups, Chung and Li proved it does. We provide
examples showing that Chung and Li's result is near-optimal in the sense that
the conclusion fails for some non-algebraic action generated by a single
homeomorphism, and for some algebraic actions of non-finitely generated abelian
groups. On the other hand, we prove that every expansive action of an amenable
group with positive entropy that has the pseudo-orbit tracing property must
admit off-diagonal asymptotic pairs.
  Using Chung and Li's algebraic characterization of expansiveness, we prove
the pseudo-orbit tracing property for a class of expansive algebraic actions.
This class includes every expansive principal algebraic action of an arbitrary
countable group.
