(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  considered the problem of whether or not there exist real numbers , which he called Z-numbers, having the property thatwhere 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 -- and A. D. Pollington , ). In particular, Pollington  proves that there are uncountably many real numbers such thatin contrast to the at most countable number of solutions of (3.4).