Some facts about continued fractions that should be better known

Abstract

In this report I will give proofs of some simple theorems concerning continued fractions that are known to the cognoscenti, but for which proofs in the literature seem to be missing, incomplete, or hard to locate. In particular, I will give two proofs of the following "folk theorem": if 5 is an irrational number whose continued fraction has bounded partial quotients, then any non-trivial linear fractional transformation of 5 also has bounded partial quotients. The second proof is of interest because it uses the connection between continued fractions and finite automata first enunciated by G. N. Raney [R]. I will assume that the reader knows basic facts about continued fractions, at the level of [HW, Chapter X].