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.
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