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.
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