Fractional parts of.

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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 allk, 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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