Theorem K implies nothing about the behavior of T on the set of
integers because it is a measure 0 subset of .
In fact, the trajectory
of any integer n can never
have the property of the conclusion of Theorem K,
for if the trajectory is eventually periodic with period k,
it cannot be uniformly distributed ,
while if it is a divergent trajectory, it cannot even be
equidistributed by (2.31).
Consequently, this connection of the problem to ergodic theory does
not seem to yield any deep insight into the problem itself.
Annotation Form Interface