Theorem H can be used to show the nonexistence of nontrivial cycles of small
period by obtaining upper bounds for the and checking that
condition (i) holds.
This approach has been taken by Crandall [28], Garner [34],
Schuppar [61] and Terras [67].
In estimating , one can show that the quantities
are never very large, so that the size of is essentially
determined by how large
can get.
The worst cases occur when is a very close approximation to ,
i.e., when is a very good rational approximation to .
The best rational approximations to are given by the convergents
of the continued fraction expansion of .
Crandall [28]
uses general properties of continued fraction convergents to obtain the
following quantitative result.