Subrecursive Graphs of Representations of Irrational Numbers.

Our goal is to compare the complexity of different representations of irrational numbers. We use the following complexity framework: one representation is subrecursive in another representation if it is possible to convert the latter into the former using an algorithm with no unbounded search. For example, the base-2 expansion is subrecursive in the Dedekind cut, but not conversely. Informally this means that the Dedekind cut provides more information and so it is generally more complex than the base-2 expansion. In the present paper, for any representation of irrational numbers, we consider the characteristic function of its graph as a new representation. The most interesting case is the graph of the continued fraction: we prove that it is strictly subrecursively between the Dedekind cut and the continued fraction itself and also that it is subrecursively incomparable to the left and right best approximations. We also prove that the graphs of the base- b sum approximations from below and from above are subrecursively equivalent to the base- b expansion and that the graphs of the left and right best approximations are subrecursively equivalent to the Dedekind cut.