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It is Riho Terras's ingenious observation that although the behavior of the total
stopping time function seems hard to analyze, a great deal can be said
about the stopping time function.
He proved the following fundamental result ([67], [68]),
also found independently by Everett [31].
Theorem A
(Terras).
The set of integers
has stopping time
has limiting asymptotic density
, i.e., the limit
exists.
In addition,
as
,
so that almost all integers have a finite stopping time.
The ideas behind Terras's analysis seem basic to a deeper understanding
of the
problem, so I describe them in detail.
In order to do this, I introduce some notation to
describe the results of the process of iterating the function
.
Given an integer n, define a sequence of 0-1 valued
quantities
by
where
.
The results of first k iterations of T are completely described by the
parity vector
since the result of k iterations is
where
and
Note that in (2.5), (2.6) both
and
are completely determined by the parity vector
given by
(2.3); I sometimes indicate this by writing
(instead of
.
The formula (2.4) shows that a necessary condition for
is that
since
is nonnegative.
Terras [67] defines the coefficient stopping time
to be the least value of k such that (2.7) holds,
and
if no such value of k exists.
It is immediate that
The function
plays an important role in the analysis of the
behavior of the stopping time function
, see Theorem C.
The formula (2.2) expresses the parity vector
as a function of n.
Terra's idea is to reverse this process and express n as a function of
.
Theorem B
The function
defined by
is periodic with period
.
The induced function
is a
permutation, and its order is a power of 2.
[Proof]
The cycle structure and order of the first few permutations
are given in Table 2.
(One-cycles are omitted.)
It is interesting to observe that the order of the permutation
seems to be much smaller than the upper bound
proved in
Theorem B.
Is there some explanation of this phenomenon?
TABLE 2. Cycle structure and order of permutation
.
Theorem B allows one to associate with each vector
of length k a unique congruence
class
given by
The integer
with
is the minimal element in
and
is the arithmetic progression:
Now I consider the relation between a vector
and stopping times for
integers
.
Define a vector
of length k
to be admissible if
- (1)
,
- (2)
, when
.
Note that all admissible vectors
of length k have
where
and
denotes the largest integer
.
The following result is due to Terras.
Theorem C
(Terras).
(a)
The set of integers with coefficient stopping time k are exactly
the set of integers in those congruence classes
for which there is an admissible vector
of length k with
.
(b) Let
for some vector
of length k.
If
is admissible, then all sufficiently large integers congruent to
have stopping time k.
If
is not admissible, then only finitely many integers
congruent to
have stopping time k.
[Proof]
Theorem C asserts that the set of integers
with a given coefficient
stopping time k is a set of arithmetic progressions
, which has the immediate consequence that
has the
asymptotic density
which is given by
Furthermore Theorem C asserts that the set
differs from
by a finite set, so that
also has an asymptotic density
which is the same as that of
.
Consequently, Theorem C implies the first part of Theorem A, that the set
of all integers with stopping time at most k have an asymptotic density
given by
where
Now the formula (2.16) can be used to prove the second part of Theorem A,
and in fact to prove the stronger result that
approaches 1 at an exponential rate as
.
Theorem D
For all
,
where
Here
is the entropy
function and
.
[Proof]
Theorem D cannot be substantially improved; it can be proved that for
any
we have
for all sufficiently large k depending on
.
Hence for any
holds for all sufficiently large k depending on
.

Contents
Next: What is the
Up: The 3x+1 problem.
Previous: A heuristic argument.