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 .