Continued fractions of primitive recursive real numbers

Abstract

A theorem, proven in the present author's Master's thesis states that a real number is ‐computable, whenever its continued fraction is in (the third Grzegorczyk class). The aim of this paper is to settle the matter with the converse of this theorem. It turns out that there exists a real number, which is ‐computable, but its continued fraction is not primitive recursive, let alone in . A question arises, whether some other natural condition on the real number can be combined with ‐computability, so that its continued fraction has low complexity. We give two such conditions. The first is ‐irrationality, based on a notion of Péter, and the second is polynomial growth of the terms of the continued fraction. Any of these two conditions, combined with ‐computability gives an (elementary) continued fraction. We conclude that all irrational algebraic real numbers and the number π have continued fractions in . All these results are generalized to higher levels of Grzegorczyk's hierarchy as well.