Subrecursive Approximations of Irrational Numbers by Variable Base Sums

Abstract

There are numerous ways to represent real numbers. We may use, e.g., Cauchy sequences, Dedekind cuts, numerical base-10 expansions, numerical base-2 expansions and continued fractions. If we work with full Turing computability, all these representations yield the same class of real numbers. If we work with some restricted notion of computability, e.g., polynomial time computability or primitive recursiveness, they do not. Irrational numbers can be represented by infinite sums of certain forms. We prove some results related to representation of irrational numbers by infinite sums.