Existence of stopping times for almost all integers.

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Several authors have investigated the range of validity of the result that

has a finite stopping time for almost all integers n by considering more general classes of periodicity linear functions. One such class

consists of all functions

which are given by

where m and d are positive integers with

and

is a fixed set of residue class representatives of the nonzero residue classes

. The 3x+1 function T is in the class

. H. Möller [54] completely characterized the functions

in the set

which have a finite stopping time for almost all integers n. He showed they are exactly those functions for which

E. Heppner [41] proved the following quantitative version of this result, thereby generalizing Theorem D.

Theorem Q

(Heppner). Let be a function in the class .

(i) If , then there exist real numbers such that for we have and as .
(ii) If , then there exist real numbers such that for we have and as .

J.-P. Allouche [1] has further sharpened Theorem Q and Matthews and Watts [51], [52] have extended it to a larger class of functions.

It is a measure of the difficulty of problems in this area that even the following apparently weak conjecture is unsolved.

EXISTENCE CONJECTURE. Let U be any function in the class . Then:

(i) U has at least one purely periodic trajectory if ;

(ii) U has at least one divergent trajectory if .

Contents Next: Fractional parts of. Up: Generalizations of problem. Previous: Algorithmic decidability questions.