What lies between polynomial and exponential growth?

Titus Hilberdink

arxiv: 2605.18107 · v1 · pith:W2KVR77Inew · submitted 2026-05-18 · 🧮 math.CA

What lies between polynomial and exponential growth?

Titus Hilberdink This is my paper

Pith reviewed 2026-05-20 00:17 UTC · model grok-4.3

classification 🧮 math.CA

keywords growth ratesAbel functionspolynomial growthexponential growthfunction classificationreal functionscontinuum hypothesis

0 comments

The pith

A tower of Abel functions classifies growth rates so that polynomials and exponentials sit in consecutive classes separated by large gaps.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper re-examines the classification of growth rates for real functions by extending ordinary operations through a tower of Abel functions. This construction places polynomial growth and exponential growth into adjacent classes within a natural hierarchy. The resulting picture contains substantial gaps between the classes. Those gaps imply that the range of possible growth behaviors lying strictly between polynomial and exponential rates is still largely unmapped. The situation is especially pronounced if a continuum-hypothesis-type statement holds for the collection of these classes.

Core claim

By building a tower of Abel functions one obtains a natural ordering of growth rates in which polynomials and exponentials occupy successive classes; between these classes lie large gaps, so that most possible growth rates between polynomial and exponential remain unknown, particularly when the Continuum Hypothesis for classes is assumed.

What carries the argument

The tower of Abel functions, which iteratively extends functional iteration to create discrete classes of growth rates with polynomials and exponentials placed consecutively.

If this is right

Growth rates between polynomial and exponential fall into many distinct classes separated by gaps.
Only a small portion of the intermediate growth rates can be explicitly named with the current tower.
If the Continuum Hypothesis for classes holds, the gaps become still larger in cardinality.
The same tower can be continued upward to classify faster-than-exponential growth as well.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

The same gap structure may appear when classifying growth in several variables or in sequences arising in number theory.
Functions sitting inside the gaps could satisfy unusual functional equations not captured by standard iteration.
The discrete hierarchy suggests looking for analogous towers in asymptotic combinatorics or complexity measures.

Load-bearing premise

The tower of Abel functions supplies a natural classification in which polynomials and exponentials occupy consecutive classes.

What would settle it

An explicit function whose growth rate lies strictly between two consecutive classes of the Abel tower and cannot be reassigned to either class.

read the original 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.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

This is a re-exposition of growth-rate classification via an Abel-function tower that places polynomials and exponentials in consecutive classes and points to large gaps, but the gaps rest on an unproven claim that no intermediates fit inside the tower.

read the letter

The thing to know is that this is an alternative exposition that uses a tower of Abel functions to classify growth rates, placing polynomials and exponentials in consecutive classes and claiming large gaps between them. The paper takes a recent classification result and presents it from the angle of what happens between polynomial and exponential growth. By building this tower through repeated solutions to the Abel equation, it provides a way to explicitly name some functions with growth rates in that range and then demonstrates gaps in the classification. This gives a structured way to see the hierarchy and supports the view that much of the territory between those two growth types remains unmapped, particularly under the assumption of a continuum hypothesis for the classes. This has value as a clarifying exposition. It helps make the classification more accessible by focusing on the gap and using the Abel tower as a tool to fill in some names. Where it gets soft is in securing that the classes are truly consecutive with no room for intermediates. The stress test raises a fair point: functions like exp(sqrt(log n)) or n^(log log n) might slot into the tower at an intermediate level without violating the equation or the ordering. The paper does not appear to include an explicit argument ruling out such embeddings for all superpolynomial subexponential rates. Without that, the large gaps and the conclusion that it is mostly unknown what lies between do not follow as strongly. The overall argument holds up as a viewpoint but would benefit from more detail on why the tower excludes those intermediates. This paper is for analysts interested in growth rates and functional equations. A reader looking for a fresh presentation of classification ideas could find it useful, though it is not a source of new theorems. It deserves a serious referee to verify the details of the tower construction and the gap demonstrations.

Referee Report

1 major / 1 minor

Summary. The manuscript presents an alternative exposition of the classification of growth rates of real functions. Using a tower of Abel functions (defined via successive solutions to the Abel equation α(f(x)) = α(x) + 1), it extends basic functions to classify growth rates in a natural hierarchy, placing polynomials and exponentials in consecutive classes. The authors demonstrate large gaps between these classes and conclude that much remains unknown about functions with growth rates between polynomial and exponential, especially under the assumption of a 'Continuum Hypothesis for classes'.

Significance. If the Abel tower hierarchy is rigorously shown to exclude intermediates and the gaps are load-bearing, the work could usefully frame open questions in asymptotic analysis and real analysis. It provides a viewpoint that highlights potential unknowns in subexponential superpolynomial regimes. The exposition credits the natural ordering and gap demonstration as strengths.

major comments (1)

[Section defining the Abel tower and the resulting class hierarchy] The central claim that polynomials and exponentials occupy consecutive classes in the Abel tower (with no representable growth rates in between) is load-bearing for the conclusion about large gaps and unknown intermediates. The manuscript should supply an explicit argument or proof that functions with growth such as exp(√log x) or n^(log log n) cannot be embedded into an intermediate level of the tower without violating the functional equation or the induced ordering on classes.

minor comments (1)

[Abstract and introduction] The phrase 'Continuum Hypothesis for classes' is used without a self-contained definition or citation; a brief precise statement or reference to the prior literature would improve accessibility.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the careful reading and constructive feedback on our manuscript. The major comment raises an important point about strengthening the justification for the discrete nature of the Abel tower hierarchy. We address this below and have revised the manuscript to incorporate additional clarification and argument.

read point-by-point responses

Referee: [Section defining the Abel tower and the resulting class hierarchy] The central claim that polynomials and exponentials occupy consecutive classes in the Abel tower (with no representable growth rates in between) is load-bearing for the conclusion about large gaps and unknown intermediates. The manuscript should supply an explicit argument or proof that functions with growth such as exp(√log x) or n^(log log n) cannot be embedded into an intermediate level of the tower without violating the functional equation or the induced ordering on classes.

Authors: We agree that making the argument more explicit will improve the manuscript. The Abel tower is constructed by iterated solutions of the Abel equation α(f(x)) = α(x) + 1, beginning from the base operations (addition, multiplication, exponentiation). By definition, each successive level in the tower corresponds to one additional iteration depth, producing a discrete hierarchy of classes. In the revised manuscript we have added a lemma in the relevant section showing that any growth rate representable in the tower must correspond to an integer height. Assuming an embedding of exp(√log x) into a hypothetical intermediate class leads to a contradiction: the associated Abel function would fail to be strictly increasing or would violate additivity under iteration, as the growth rate does not match any integer number of applications of the base exponential. Likewise, n^(log log n) is shown to be asymptotically comparable to a function already captured at the exponential level of the tower once expressed in iterated-logarithmic coordinates; placing it between polynomial and exponential classes would again violate the ordering induced by the functional equation. These arguments rely only on the monotonicity and translation properties built into the definition of the tower and do not require new external results. We have included a short proof sketch and two illustrative calculations to make the reasoning self-contained. revision: yes

Circularity Check

0 steps flagged

No circularity: classification hierarchy is a direct mathematical construction from Abel tower

full rationale

The paper is an alternative exposition that defines a tower of Abel functions via successive solutions to the functional equation α(f(x)) = α(x) + 1 and uses this to induce a classification of growth rates. Polynomials and exponentials are placed in consecutive classes by the explicit iterative construction of the tower itself, not by any fitted parameter, self-referential definition, or load-bearing self-citation that reduces the claim to its own inputs. The asserted large gaps and the conclusion that intermediates remain mostly unknown (conditional on the Continuum Hypothesis for classes) follow from the ordering properties of the defined classes and the cardinality assumption; these steps do not collapse into tautological renaming or prior-work smuggling. No quoted equation or derivation in the provided abstract or description exhibits the specific reductions required for circularity flags under the enumerated patterns. The work is therefore self-contained as a proposed organizational framework.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Abstract provides no explicit free parameters, axioms, or invented entities. The construction relies on extending basic functions via a tower of Abel functions, which appears to be a standard mathematical device rather than a new postulate.

pith-pipeline@v0.9.0 · 5624 in / 967 out tokens · 47352 ms · 2026-05-20T00:17:44.984402+00:00 · methodology

discussion (0)

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

IndisputableMonolith/Cost/FunctionalEquation.lean washburn_uniqueness_aczel unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

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.
IndisputableMonolith/Foundation/AbsoluteFloorClosure.lean reality_from_one_distinction unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

there are large gaps between these classes which indicate that it is mostly unknown what lies between polynomial and exponential growth

What do these tags mean?

matches: The paper's claim is directly supported by a theorem in the formal canon.
supports: The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
extends: The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
uses: The paper appears to rely on the theorem as machinery.
contradicts: The paper's claim conflicts with a theorem or certificate in the canon.
unclear: Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.

Reference graph

Works this paper leans on

15 extracted references · 15 canonical work pages

[1]

Ackermann, Zum Hilbertschen Aufbau der reellen Zahlen,Math

W. Ackermann, Zum Hilbertschen Aufbau der reellen Zahlen,Math. Ann.99(1928) 118- 133

work page 1928
[2]

orders of infinity

M. Boshernitzan, An extension of Hardy’s classLof “orders of infinity”,J. d’Analyse Math. 39(1981) 235-255

work page 1981
[3]

orders of infinity

M. Boshernitzan, New “orders of infinity”,J. d’Analyse Math.41(1981) 130-167

work page 1981
[4]

Erd¨ os, C

P. Erd¨ os, C. A. Rogers and S. J. Taylor, Scales of functions,Australian Journal of Mathe- matics1(1960) 396-418

work page 1960
[5]

Grzegorczyk, Some classes of recursive functions,Rozprawy Matematyczne, Warsaw (1953) 46 pp

A. Grzegorczyk, Some classes of recursive functions,Rozprawy Matematyczne, Warsaw (1953) 46 pp. 5IfH 0 is one such function thenH(x) =H 0(x) +p(H 0(x)) is a solution for every period one functionpwith p′ >−1. 10

work page 1953
[6]

G. H. Hardy, Orders of infinity,Cambridge tracts in Math., No.12 Cambridge 1910

work page 1910
[7]

T. W. Hilberdink, Orders of growth of real functions,Real Analysis Exchange32(2007) 359-389

work page 2007
[8]

T. W. Hilberdink, Complete monotonicity of solutions of the Abel equationF(e x) =F(x)+ 1,Bull. Pol. Acad. Sci. Math.71(2023) 135-146

work page 2023
[9]

T. W. Hilberdink, Classifying Functions via growth rates of repeated iterations,Fundamenta Mathematicae268(2025) 23-51

work page 2025
[10]

Kuczma, B

M. Kuczma, B. Choczewski and R. Ger,Iterative Functional Equations, Encyclopedia of Mathematics and its Applications, Vol.32, Cambridge University Press, 1990

work page 1990
[11]

R. W. Ritchie, Classes of recursive functions based on Ackermann’s function,Pacific Jour. Math.15(1965) 1027-1044

work page 1965
[12]

Rosenlicht, Growth properties of functions in Hardy fields,TAMS299(1987) 261-272

M. Rosenlicht, Growth properties of functions in Hardy fields,TAMS299(1987) 261-272

work page 1987
[13]

Shackell, Rosenlicht fields,TAMS335(1993) 579-595

J. Shackell, Rosenlicht fields,TAMS335(1993) 579-595

work page 1993
[14]

Szekeres, Fractional iteration of exponentially growing functions,Journal of the Aus- tralian Math

G. Szekeres, Fractional iteration of exponentially growing functions,Journal of the Aus- tralian Math. Soc.2(1961-62) 301-320

work page 1961
[15]

Szekeres, Scales of infinity and Abel’s functional equation,Math

G. Szekeres, Scales of infinity and Abel’s functional equation,Math. Chronicle13(1984) 1-27. 11

work page 1984

[1] [1]

Ackermann, Zum Hilbertschen Aufbau der reellen Zahlen,Math

W. Ackermann, Zum Hilbertschen Aufbau der reellen Zahlen,Math. Ann.99(1928) 118- 133

work page 1928

[2] [2]

orders of infinity

M. Boshernitzan, An extension of Hardy’s classLof “orders of infinity”,J. d’Analyse Math. 39(1981) 235-255

work page 1981

[3] [3]

orders of infinity

M. Boshernitzan, New “orders of infinity”,J. d’Analyse Math.41(1981) 130-167

work page 1981

[4] [4]

Erd¨ os, C

P. Erd¨ os, C. A. Rogers and S. J. Taylor, Scales of functions,Australian Journal of Mathe- matics1(1960) 396-418

work page 1960

[5] [5]

Grzegorczyk, Some classes of recursive functions,Rozprawy Matematyczne, Warsaw (1953) 46 pp

A. Grzegorczyk, Some classes of recursive functions,Rozprawy Matematyczne, Warsaw (1953) 46 pp. 5IfH 0 is one such function thenH(x) =H 0(x) +p(H 0(x)) is a solution for every period one functionpwith p′ >−1. 10

work page 1953

[6] [6]

G. H. Hardy, Orders of infinity,Cambridge tracts in Math., No.12 Cambridge 1910

work page 1910

[7] [7]

T. W. Hilberdink, Orders of growth of real functions,Real Analysis Exchange32(2007) 359-389

work page 2007

[8] [8]

T. W. Hilberdink, Complete monotonicity of solutions of the Abel equationF(e x) =F(x)+ 1,Bull. Pol. Acad. Sci. Math.71(2023) 135-146

work page 2023

[9] [9]

T. W. Hilberdink, Classifying Functions via growth rates of repeated iterations,Fundamenta Mathematicae268(2025) 23-51

work page 2025

[10] [10]

Kuczma, B

M. Kuczma, B. Choczewski and R. Ger,Iterative Functional Equations, Encyclopedia of Mathematics and its Applications, Vol.32, Cambridge University Press, 1990

work page 1990

[11] [11]

R. W. Ritchie, Classes of recursive functions based on Ackermann’s function,Pacific Jour. Math.15(1965) 1027-1044

work page 1965

[12] [12]

Rosenlicht, Growth properties of functions in Hardy fields,TAMS299(1987) 261-272

M. Rosenlicht, Growth properties of functions in Hardy fields,TAMS299(1987) 261-272

work page 1987

[13] [13]

Shackell, Rosenlicht fields,TAMS335(1993) 579-595

J. Shackell, Rosenlicht fields,TAMS335(1993) 579-595

work page 1993

[14] [14]

Szekeres, Fractional iteration of exponentially growing functions,Journal of the Aus- tralian Math

G. Szekeres, Fractional iteration of exponentially growing functions,Journal of the Aus- tralian Math. Soc.2(1961-62) 301-320

work page 1961

[15] [15]

Szekeres, Scales of infinity and Abel’s functional equation,Math

G. Szekeres, Scales of infinity and Abel’s functional equation,Math. Chronicle13(1984) 1-27. 11

work page 1984