What lies between polynomial and exponential growth?

TL;DR

This paper explores the classification of growth rates of real functions between polynomial and exponential, introducing a new framework using Abel functions to understand the gaps and unknowns in this range.

Contribution

It presents an alternative approach to classifying intermediate growth rates using a tower of Abel functions, highlighting large gaps and the potential impact of the Continuum Hypothesis.

Findings

Large gaps exist between polynomial and exponential growth classes.

The Abel function tower provides a natural classification framework.

Uncertainty remains about the functions lying between polynomial and exponential growth.

Abstract

In this paper we give an alternative exposition of a recent paper regarding the classification of growth rates of real functions. We take a different point of view, focussing on understanding possible growth rates between polynomial and exponential. In order to be able to explicitly name a range of such functions, we first need to extend our basic functions. We do this via a 'tower' of Abel functions. With these one can classify functions in a natural way with polynomials and exponentials in consecutive classes. We show there are large gaps between these classes which indicate that it is mostly unknown what lies between polynomial and exponential growth, especially if the "Continuum Hypothesis for classes" is true.