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  considered the problem of whether or
not there exist real numbers , which he called
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
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
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 allk,
unlike the behavior of the function .