(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).