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Attempts to understand the distribution
of the sequence have uncovered oblique
connections with ergodic-theoretic aspects of a generalization of the
**3x+1** problem.
It is conjectured that the sequence is uniformly distributed
.

(This conjecture seems intractable at present.)

One approach to this problem is to determine what kinds of
distributions can occur for sequences ,
where is a fixed real number.
In this vein K. Mahler [49] considered the problem of whether or
not there exist real numbers , which he called
* Z-numbers*,
having the property that

where is the fractional part of **x**.
He showed that the set of **Z**-numbers is countable, by showing that there
is at most one **Z**-number in each interval
, for .
He went on to show that a necessary condition for the existence of a **Z**-number
in the interval is that the trajectory
of **n** produced by the
periodically linear function
satisfy
Mahler concluded from this that is unlikely that any **Z**-numbers exist.
This is supported by the following heuristic argument.
The function **W** may be interpreted as acting on the 2-adic
integers by (3.5), and it has properties
exactly analogous to the properties of **T** given by Theorem K.
In particular, for almost all 2-adic integers the sequence of
iterates
has infinitely many values **k** with .
Thus if a given behaves like almost all 2-adic integers
, then (3.6) will not hold for **n**.
Note that it is possible that all the trajectories
for are uniformly
distributed for * all* **k**,
unlike the behavior of the function .

In passing, I note that the possible distributions of
;
for real have an intricate structure
(see G. Choquet [16]--[22]
and A. D. Pollington [57], [58]).
In particular, Pollington [58] proves that there are
uncountably many real numbers such that

in contrast to the at most countable number of solutions
of (3.4).

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** Next:** Conclusion.
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** Previous:** Existence of stopping