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 C

The assertions made in (a) about coefficient stopping times follow from the definition of admissibility, because that definition asserts that

(i) ,

(ii) for .

To prove (b), first note that if is admissible of length k, then

and so all elements of have stopping time at least k. Now define by


where , and note that (2.11) implies that

for all admissible . Now for for an admissible , (2.4) may be rewritten as


Hence when is admissible, those n in with


have stopping time k, and in this case.

Now suppose is not admissible. There are two cases, depending on whether or not some initial segment of is admissible. No initial segment of is admissible if and only if


and when (2.15) holds say that is inflating. If is inflating, so that for all n in by (2.4), so that no elements of have stopping time k or less. In the remaining case has an initial segment with i < k -1 which is admissible. Now and all sufficiently large elements of have stopping time i + 1 < k by the argument just given.