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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