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.