Subrecursive Approximations of Irrational Numbers by Variable Base Sums

TL;DR

This paper explores how irrational numbers can be represented through specific infinite sums, focusing on subrecursive approximations within restricted computational frameworks.

Contribution

It introduces new results on representing irrationals using variable base sums under subrecursive computability constraints.

Findings

Irrational numbers can be approximated by variable base sums.

Representation results depend on the computational restrictions.

The paper extends understanding of real number representations in subrecursive contexts.

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.